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| In [[mathematics]], the [[classification of finite simple groups]] states that every finite [[simple group]] is [[cyclic group|cyclic]], or [[alternating group|alternating]], or in one of 16 families of [[groups of Lie type]], or one of 26 [[sporadic group]]s.
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| The list below gives all finite simple groups, together with their [[order (group theory)|order]], the size of the [[Schur multiplier]], the size of the [[outer automorphism group]], usually some small [[group representation|representations]], and lists of all duplicates. (In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group ''B<sub>n</sub>''(''q'') has the same order as ''C<sub>n</sub>''(''q'') for ''q'' odd, ''n'' > 2; and the groups A<sub>8</sub> = ''A''<sub>3</sub>(2) and ''A''<sub>2</sub>(4) both have orders 20160.)
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| '''Notation:''' ''n'' is a positive integer, ''q'' > 1 is a power of a prime number ''p'', and is the order of some underlying [[finite field]]. The order of the outer automorphism group is written as ''d''·''f''·''g'', where ''d'' is the order of the group of "diagonal automorphisms", ''f'' is the order of the (cyclic) group of "field automorphisms" (generated by a [[Frobenius automorphism]]), and ''g'' is the order of the group of "graph automorphisms" (coming from automorphisms of the [[Dynkin diagram]]).
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| ==Infinite families==
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| ===[[Cyclic group]]s ''Z<sub>p</sub>''===
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| '''Simplicity:''' Simple for ''p'' a prime number.
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| '''Order:''' ''p''
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| '''Schur multiplier:''' Trivial.
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| '''Outer automorphism group:''' Cyclic of order ''p'' − 1.
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| '''Other names:''' ''Z/pZ''
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| '''Remarks:''' These are the only simple groups that are not [[perfect group|perfect]].
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| ===A<sub>''n''</sub>, ''n'' > 4, [[Alternating group]]s===
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| <!-- Note that this page uses italic for group of Lie type A, but Roman for the alternating groups.-->
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| '''Simplicity:''' Solvable for ''n'' < 5, otherwise simple.
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| '''Order:''' ''n''!/2 when ''n'' > 1.
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| '''Schur multiplier:''' 2 for ''n'' = 5 or ''n'' > 7, 6 for ''n'' = 6 or 7; see [[Covering groups of the alternating and symmetric groups]]
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| '''Outer automorphism group:''' In general 2. Exceptions: for ''n'' = 1, ''n'' = 2, it is trivial, and for [[outer automorphism group#The outer automorphisms of the symmetric groups|''n'' = 6]], it has order 4 (elementary abelian).
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| '''Other names:''' ''Alt<sub>n</sub>''.
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| There is an unfortunate conflict with the notation for the (unrelated) groups ''A<sub>n</sub>''(''q''), and some authors use various different fonts for A<sub>''n''</sub> to distinguish them. In particular,
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| in this article we make the distinction by setting the alternating groups A<sub>''n''</sub> in Roman font and the Lie-type groups ''A<sub>n</sub>''(''q'') in italic.
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| '''Isomorphisms:''' A<sub>1</sub> and A<sub>2</sub> are trivial. A<sub>3</sub> is cyclic of order 3. A<sub>4</sub> is isomorphic to ''A''<sub>1</sub>(3) (solvable). A<sub>5</sub> is isomorphic to ''A''<sub>1</sub>(4) and to ''A''<sub>1</sub>(5). A<sub>6</sub> is isomorphic to ''A''<sub>1</sub>(9) and to the derived group ''B''<sub>2</sub>(2)'. A<sub>8</sub> is isomorphic to ''A''<sub>3</sub>(2).
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| '''Remarks:''' An [[Index of a subgroup|index]] 2 subgroup of the [[symmetric group]] of permutations of ''n'' points when ''n'' > 1.
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| === [[Chevalley group]]s ''A<sub>n</sub>''(''q''), ''B<sub>n</sub>''(''q'') ''n'' > 1, ''C<sub>n</sub>''(''q'') ''n'' > 2, ''D<sub>n</sub>''(''q'') ''n'' > 3 === | |
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| {| class="wikitable"
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| |width="15%"|
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| !width="21%"| [[Chevalley group]]s ''A<sub>n</sub>''(''q'') <br>linear groups
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| !width="21%"| [[Chevalley group]]s ''B<sub>n</sub>''(''q'') ''n'' > 1<br>[[orthogonal groups]]
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| !width="21%"| [[Chevalley group]]s ''C<sub>n</sub>''(''q'') ''n'' > 2<br>[[symplectic group]]s
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| !width="22%"| [[Chevalley group]]s ''D<sub>n</sub>''(''q'') ''n'' > 3<br>[[orthogonal group]]s
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| |-
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| !valign=top style="text-align: right;"| ''Simplicity:''
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| |valign=top| ''A''<sub>1</sub>(2) and ''A''<sub>1</sub>(3) are solvable, the others are simple.
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| |valign=top| ''B''<sub>2</sub>(2) is not simple but its derived group ''B''<sub>2</sub>(2)′ is a simple subgroup of index 2; the others are simple.
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| |valign=top| All simple
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| |valign=top| All simple
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| |-
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| !valign=top style="text-align: right;"| ''Order:''
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| |valign=top| <math>{1\over (n+1,q-1)}q^{n(n+1)/2}\prod_{i=1}^n(q^{i+1}-1)</math>
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| |valign=top| <math>{1\over (2,q-1)}q^{n^2}\prod_{i=1}^n(q^{2i}-1)</math>
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| |valign=top| <math>{1\over (2,q-1)}q^{n^2}\prod_{i=1}^n(q^{2i}-1)</math>
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| |valign=top| <math>{1\over (4,q^n-1)}q^{n(n-1)}(q^n-1)\prod_{i=1}^{n-1}(q^{2i}-1)</math>
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| |-
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| !valign=top style="text-align: right;"| ''Schur multiplier:''
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| |valign=top| For the simple groups it is cyclic of order (''n''+1, ''q'' − 1) except for ''A''<sub>1</sub>(4) (order 2), ''A''<sub>1</sub>(9) (order 6), ''A''<sub>2</sub>(2) (order 2), ''A''<sub>2</sub>(4) (order 48, product of cyclic groups of orders 3, 4, 4), ''A''<sub>3</sub>(2) (order 2).
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| |valign=top| (2,''q'' − 1) except for ''B''<sub>2</sub>(2) = S<sub>6</sub> (order 2 for ''B''<sub>2</sub>(2), order 6 for ''B''<sub>2</sub>(2)′) and ''B''<sub>3</sub>(2) (order 2) and ''B''<sub>3</sub>(3) (order 6).
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| |valign=top| (2,''q'' − 1) except for ''C<sub>3</sub>''(2) (order 2).
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| |valign=top| The order is (4, ''q<sup>n</sup>'' − 1) (cyclic for ''n'' odd, elementary abelian for ''n'' even) except for ''D''<sub>4</sub>(2) (order 4, elementary abelian).
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| !valign=top style="text-align: right;"| ''Outer automorphism group:''
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| |valign=top| (2, ''q'' − 1) ·''f''·1 for ''n'' = 1; (''n''+1, ''q'' − 1) ·''f''·2 for ''n'' > 1, where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| (2, ''q'' − 1) ·''f''·1 for ''q'' odd or ''n''>2; (2, ''q'' − 1) ·''f''·2 if ''q'' is even and ''n''=2, where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| (2, ''q'' − 1) ·''f''·1 where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| (2, ''q'' − 1) <sup>2</sup>·''f''·''S''<sub>3</sub> for ''n''=4, (2, ''q'' − 1) <sup>2</sup>·''f''·2 for ''n>4'' even, (4, ''q<sup>n</sup>'' − 1)·''f''·2 for ''n'' odd, where ''q'' = ''p<sup>f</sup>'', and ''S''<sub>3</sub> is the symmetric group of order 3! on 3 points.
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| !valign=top style="text-align: right;"| ''Other names:''
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| |valign=top| [[Projective special linear group]]s, ''PSL<sub>n+1</sub>(q)'', ''L''<sub>''n''+1</sub>(''q''), ''PSL''(''n''+1,''q'')
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| |valign=top| ''O''<sub>2''n''+1</sub>(''q''), Ω<sub>2''n''+1</sub>(''q'') (for ''q'' odd).
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| |valign=top| Projective symplectic group, ''PSp''<sub>2''n''</sub>(''q''), ''PSp''<sub>''n''</sub>(''q'') (not recommended), ''S''<sub>2''n''</sub>(''q''), Abelian group (archaic).
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| |valign=top| ''O''<sub>2''n''</sub><sup>+</sup>(''q''), ''PΩ''<sub>2''n''</sub><sup>+</sup>(''q''). "[[Orthogonal group|Hypoabelian group]]" is an archaic name for this group in characteristic 2.
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| !valign=top style="text-align: right;"| ''Isomorphisms:''
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| |valign=top| ''A''<sub>1</sub>(2) is isomorphic to the symmetric group on 3 points of order 6. ''A''<sub>1</sub>(3) is isomorphic to the alternating group A<sub>4</sub> (solvable). ''A''<sub>1</sub>(4) and ''A''<sub>1</sub>(5) are isomorphic, and are both isomorphic to the alternating group A<sub>5</sub>. ''A''<sub>1</sub>(7) and ''A''<sub>2</sub>(2) are isomorphic. ''A''<sub>1</sub>(8) is isomorphic to the derived group <sup>2</sup>''G''<sub>2</sub>(3)′. ''A''<sub>1</sub>(9) is isomorphic to A<sub>6</sub> and to the derived group ''B''<sub>2</sub>(2)′. ''A''<sub>3</sub>(2) is isomorphic to A<sub>8</sub>.
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| |valign=top| ''B<sub>n</sub>''(2<sup>''m''</sup>) is isomorphic to ''C<sub>n</sub>''(2<sup>''m''</sup>). ''B''<sub>2</sub>(2) is isomorphic to the symmetric group on 6 points, and the derived group ''B''<sub>2</sub>(2)′ is isomorphic to ''A''<sub>1</sub>(9) and to A<sub>6</sub>. ''B''<sub>2</sub>(3) is isomorphic to <sup>2</sup>''A''<sub>3</sub>(2<sup>2</sup>).
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| |valign=top| ''C<sub>n</sub>''(2<sup>''m''</sup>) is isomorphic to ''B<sub>n</sub>''(2<sup>''m''</sup>)
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| |valign=top|
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| !valign=top style="text-align: right;"| ''Remarks:''
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| |valign=top| These groups are obtained from the [[general linear group]]s ''GL''<sub>''n''+1</sub>(''q'') by taking the elements of determinant 1 (giving the [[special linear group]]s ''SL''<sub>''n''+1</sub>(''q'')) and then ''quotienting out'' by the center.
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| |valign=top| This is the group obtained from the [[orthogonal group]] in dimension 2''n''+1 by taking the kernel of the determinant and [[spinor norm]] maps. ''B<sub>1</sub>''(''q'') also exists, but is the same as ''A<sub>1</sub>''(''q''). ''B<sub>2</sub>''(''q'') has a non-trivial graph automorphism when ''q'' is a power of 2.
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| |valign=top| This group is obtained from the [[symplectic group]] in 2''n'' dimensions by ''quotienting out'' the center. ''C''<sub>1</sub>(''q'') also exists, but is the same as ''A''<sub>1</sub>(''q''). ''C''<sub>2</sub>(''q'') also exists, but is the same as ''B''<sub>2</sub>(''q'').
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| |valign=top| This is the group obtained from the [[split orthogonal group]] in dimension 2''n'' by taking the kernel of the determinant (or [[Orthogonal group|Dickson invariant]] in characteristic 2) and [[spinor norm]] maps and then killing the center. The groups of type ''D''<sub>4</sub> have an unusually large diagram automorphism group of order 6, containing the [[triality]] automorphism. ''D''<sub>2</sub>(''q'') also exists, but is the same as ''A''<sub>1</sub>(''q'')×''A''<sub>1</sub>(''q''). ''D''<sub>3</sub>(''q'') also exists, but is the same as ''A''<sub>3</sub>(''q'').
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| |}
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| === [[Chevalley group]]s ''E''<sub>6</sub>(''q''), ''E''<sub>7</sub>(''q''), ''E''<sub>8</sub>(''q''), ''F''<sub>4</sub>(''q''), ''G''<sub>2</sub>(''q'') ===
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| {| class="wikitable"
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| |width="15%"|
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| !width="17%"| [[Chevalley group]]s ''E''<sub>6</sub>(''q'')
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| !width="17%"| [[Chevalley group]]s ''E''<sub>7</sub>(''q'')
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| !width="17%"| [[Chevalley group]]s ''E''<sub>8</sub>(''q'')
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| !width="17%"| [[Chevalley group]]s ''F''<sub>4</sub>(''q'')
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| !width="17%"| [[Chevalley group]]s ''G''<sub>2</sub>(''q'')
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| ! style="text-align: right;"| ''Simplicity:''
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| |valign=top| All simple
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| |valign=top| All simple
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| |valign=top| All simple
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| |valign=top| All simple
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| |valign=top| ''G''<sub>2</sub>(2) is not simple but its derived group ''G''<sub>2</sub>(2)′ is a simple subgroup of index 2; the others are simple.
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| ! style="text-align: right;"| ''Order:''
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| |valign=top| ''q''<sup>36</sup>(''q''<sup>12</sup> − 1)(''q''<sup>9</sup> − 1)(''q''<sup>8</sup> − 1)(''q''<sup>6</sup> − 1)(''q''<sup>5</sup> − 1)(''q''<sup>2</sup> − 1)/(3,''q'' − 1)
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| |valign=top| ''q''<sup>63</sup>(''q''<sup>18</sup> − 1)(''q''<sup>14</sup> − 1)(''q''<sup>12</sup> − 1)(''q''<sup>10</sup> − 1)(''q''<sup>8</sup> − 1)(''q''<sup>6</sup> − 1)(''q''<sup>2</sup> − 1)/(2,''q'' − 1)
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| |valign=top| ''q''<sup>120</sup>(''q''<sup>30</sup>−1)(''q''<sup>24</sup>−1)(''q''<sup>20</sup>−1)(''q''<sup>18</sup>−1)(''q''<sup>14</sup>−1)(''q''<sup>12</sup>−1)(''q''<sup>8</sup>−1)(''q''<sup>2</sup>−1)
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| |valign=top| ''q''<sup>24</sup>(''q''<sup>12</sup>−1)(''q''<sup>8</sup>−1)(''q''<sup>6</sup>−1)(''q''<sup>2</sup>−1)
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| |valign=top| ''q''<sup>6</sup>(''q''<sup>6</sup>−1)(''q''<sup>2</sup>−1)
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| ! style="text-align: right;"| ''Schur multiplier:''
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| |valign=top| (3,''q'' − 1)
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| |valign=top| (2,''q'' − 1)
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| |valign=top| Trivial
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| |valign=top| Trivial except for ''F''<sub>4</sub>(2) (order 2).
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| |valign=top| Trivial for the simple groups except for ''G''<sub>2</sub>(3) (order 3) and ''G''<sub>2</sub>(4) (order 2).
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| ! style="text-align: right;"| ''Outer automorphism group:''
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| |valign=top| (3, ''q'' − 1) ·''f''·2 where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| (2, ''q'' − 1) ·''f''·1 where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| 1·''f''·1 where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| 1·''f''·1 for ''q'' odd, 1·''f''·2 for ''q'' even, where ''q'' = ''p<sup>f</sup>''.
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| |valign=top| 1·''f''·1 for ''q'' not a power of 3, 1·''f''·2 for ''q'' a power of 3, where ''q'' = ''p<sup>f</sup>''.
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| ! style="text-align: right;"| ''Other names:''
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| |valign=top| Exceptional Chevalley group
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| |valign=top| Exceptional Chevalley group
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| |valign=top| Exceptional Chevalley group
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| |valign=top| Exceptional Chevalley group
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| |valign=top| Exceptional Chevalley group
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| |-
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| ! style="text-align: right;"| ''Isomorphisms:''
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| |valign=top|
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| |valign=top|
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| |valign=top|
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| |valign=top|
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| |valign=top| The derived group ''G''<sub>2</sub>(2)′ is isomorphic to <sup>2</sup>''A''<sub>2</sub>(3<sup>2</sup>).
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| |-
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| ! style="text-align: right;"| ''Remarks:''
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| |valign=top|Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.
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| |valign=top|Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.
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| |valign=top|It acts on the corresponding Lie algebra of dimension 248. ''E''<sub>8</sub>(3) contains the Thompson simple group.
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| |valign=top|These groups act on 27 dimensional exceptional [[Jordan algebra]]s, which gives them 26 dimensional representations. They also act on the corresponding Lie algebras of dimension 52. ''F''<sub>4</sub>(''q'') has a non-trivial graph automorphism when ''q'' is a power of 2.
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| |valign=top|These groups are the automorphism groups of 8-dimensional [[Cayley algebra]]s over finite fields, which gives them 7 dimensional representations. They also act on the corresponding Lie algebras of dimension 14. ''G''<sub>2</sub>(''q'') has a non-trivial graph automorphism when ''q'' is a power of 3.
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| |}
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| === [[Group of Lie type#Steinberg_groups|Steinberg group]]s <sup>2</sup>''A<sub>n</sub>''(''q''<sup>2</sup>) ''n'' > 1, <sup>2</sup>''D<sub>n</sub>''(''q''<sup>2</sup>) ''n'' > 3, <sup>2</sup>E<sub>6</sub>''(''q''<sup>2</sup>), <sup>3</sup>''D''<sub>4</sub>(''q''<sup>3</sup>) ===
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| {| class="wikitable"
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| !valign=top width="21%"| [[Group of Lie type#Steinberg_groups|Steinberg group]]s <sup>2</sup>''A<sub>n</sub>''(''q''<sup>2</sup>) ''n'' > 1<br>[[unitary group]]s
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| !valign=top width="21%"| [[Group of Lie type#Steinberg_groups|Steinberg group]]s <sup>2</sup>''D<sub>n</sub>''(''q''<sup>2</sup>) ''n'' > 3<br>[[orthogonal group]]s
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| !valign=top width="21%"| [[Group of Lie type#Steinberg_groups|Steinberg group]]s <sup>2</sup>E<sub>6</sub>''(''q''<sup>2</sup>)
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| !valign=top width="22%"| [[³D₄|Steinberg group]]s <sup>3</sup>''D''<sub>4</sub>(''q''<sup>3</sup>)
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| ! style="text-align: right;"| ''Simplicity:''
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| |valign=top| <sup>2</sup>''A''<sub>2</sub>(2<sup>2</sup>) is solvable, the others are simple.
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| |valign=top| All simple
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| |valign=top| All simple
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| |valign=top| All simple
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| |-
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| ! style="text-align: right;"| ''Order:''
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| |valign=top| <math>{1\over (n+1,q+1)}q^{n(n+1)/2}\prod_{i=1}^n(q^{i+1}-(-1)^{i+1})</math>
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| |valign=top| <math>{1\over (4,q^n+1)}q^{n(n-1)}(q^n+1)\prod_{i=1}^{n-1}(q^{2i}-1)</math>
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| |valign=top| ''q''<sup>36</sup>(''q''<sup>12</sup>−1)(''q''<sup>9</sup>+1)(''q''<sup>8</sup>−1)(''q''<sup>6</sup>−1)(''q''<sup>5</sup>+1)(''q''<sup>2</sup>−1)/(3,''q''+1)
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| |valign=top| ''q''<sup>12</sup>(''q''<sup>8</sup>+''q''<sup>4</sup>+1)(''q''<sup>6</sup>−1)(''q''<sup>2</sup>−1)
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| |-
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| !valign=top style="text-align: right;"| ''Schur multiplier:''
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| |valign=top| Cyclic of order (''n'' + 1, ''q'' + 1) for the simple groups, except for <sup>2</sup>''A''<sub>3</sub>(2<sup>2</sup>) (order 2), <sup>2</sup>''A''<sub>3</sub>(3<sup>2</sup>) (order 36, product of cyclic groups of orders 3,3,4), <sup>2</sup>''A''<sub>5</sub>(2<sup>2</sup>) (order 12, product of cyclic groups of orders 2,2,3)
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| |valign=top| Cyclic of order (4, ''q<sup>n</sup>'' + 1)
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| |valign=top| (3, ''q'' + 1) except for ''<sup>2</sup>E<sub>6</sub>''(2<sup>2</sup>) (order 12, product of cyclic groups of orders 2,2,3).
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| |valign=top| Trivial
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| |-
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| !valign=top style="text-align: right;"| ''Outer automorphism group:''
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| |valign=top| (''n''+1, ''q'' + 1) ·''f''·1 where ''q''<sup>2</sup> = ''p<sup>f</sup>''
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| |valign=top| (4, ''q<sup>n</sup>'' + 1) ·''f''·1 where ''q''<sup>2</sup> = ''p<sup>f</sup>''
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| |valign=top| (3, ''q'' + 1) ·''f''·1 where ''q''<sup>2</sup> = ''p<sup>f</sup>''.
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| |valign=top| 1·''f''·1 where ''q''<sup>3</sup> = ''p<sup>f</sup>''.
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| |-
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| !valign=top style="text-align: right;"| ''Other names:''
| |
| |valign=top| Twisted Chevalley group, projective special unitary group, ''PSU''<sub>''n''+1</sub>(''q''), ''PSU''(''n''+1, ''q''), ''U''<sub>''n''+1</sub>(''q''), <sup>2</sup> ''A<sub>n</sub>''(''q''), <sup>2</sup>''A<sub>n</sub>''(''q'', ''q''<sup>2</sup>)
| |
| |valign=top| <sup>2</sup>''D<sub>n</sub>''(''q''), ''O''<sub>2''n''</sub><sup>−</sup>(''q''), ''PΩ''<sub>2''n''</sub><sup>−</sup>(''q''), twisted Chevalley group. "Hypoabelian group" is an archaic name for this group in characteristic 2.
| |
| |valign=top| ''<sup>2</sup>E<sub>6</sub>''(''q''), twisted Chevalley group.
| |
| |valign=top| <sup>3</sup>''D''<sub>4</sub>(''q''), ''D''<sub>4</sub><sup>2</sup>(''q''<sup>3</sup>)<!--Steinberg's name for it, very common in early 70s, TeX is D_4{}^2(q^3), note the horizontal space before the sup 2 -->, Twisted Chevalley groups.
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Isomorphisms:'' | |
| |valign=top| The solvable group <sup>2</sup>''A''<sub>2</sub>(2<sup>2</sup>) is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. <sup>2</sup>''A''<sub>2</sub>(3<sup>2</sup>) is isomorphic to the derived group ''G''<sub>2</sub>(2)′. <sup>2</sup>''A''<sub>3</sub>(2<sup>2</sup>) is isomorphic to ''B''<sub>2</sub>(3).
| |
| |valign=top|
| |
| |valign=top|
| |
| |valign=top|
| |
| |-
| |
| ! style="text-align: right;"| ''Remarks:''
| |
| |valign=top| This is obtained from the [[unitary group]] in ''n''+1 dimensions by taking the subgroup of elements of determinant 1 and then ''quotienting'' out by the center.
| |
| |valign=top| This is the group obtained from the non-split orthogonal group in dimension 2''n'' by taking the kernel of the determinant (or [[Orthogonal group|Dickson invariant]] in characteristic 2) and [[spinor norm]] maps and then killing the center. <sup>2</sup>''D''<sub>2</sub>(''q''<sup>2</sup>) also exists, but is the same as ''A''<sub>1</sub>(''q''<sup>2</sup>). <sup>2</sup>''D''<sub>3</sub>(''q''<sup>2</sup>) also exists, but is the same as <sup>2</sup>''A''<sub>3</sub>(''q''<sup>2</sup>).
| |
| |valign=top| One of the exceptional double covers of ''<sup>2</sup>E<sub>6</sub>''(2<sup>2</sup>) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.
| |
| |valign=top| ''<sup>3</sup>D<sub>4</sub>''(2<sup>3</sup>) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.
| |
| |}
| |
| | |
| ===<sup>2</sup>''B''<sub>2</sub>(2<sup>2''n''+1</sup>) [[Suzuki groups]]===
| |
| | |
| '''Simplicity:''' Simple for ''n'' ≥ 1. The group
| |
| <sup>2</sup>''B''<sub>2</sub>(2) is solvable.
| |
| | |
| '''Order:'''
| |
| ''q''<sup>2</sup>
| |
| (''q''<sup>2</sup> + 1)
| |
| (''q'' − 1)
| |
| where
| |
| ''q'' = 2<sup>2''n''+1</sup>.
| |
| | |
| '''Schur multiplier:''' Trivial for ''n'' ≠ 1, elementary abelian of order 4
| |
| for <sup>2</sup>''B''<sub>2</sub>(8).
| |
| | |
| '''Outer automorphism group:'''
| |
| : 1·''f''·1
| |
| where ''f'' = 2''n'' + 1.
| |
| | |
| '''Other names:''' Suz(2<sup>2''n''+1</sup>), Sz(2<sup>2''n''+1</sup>).
| |
| | |
| '''Isomorphisms:''' <sup>2</sup>''B''<sub>2</sub>(2) is the Frobenius group of order 20.
| |
| | |
| '''Remarks:''' Suzuki group are [[Zassenhaus group]]s acting on sets of size (2<sup>2''n''+1</sup>)<sup>2</sup> + 1, and have 4 dimensional representations over the field with 2<sup>2''n''+1</sup> elements. They are the only non-cyclic simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group.
| |
| | |
| ===<sup>2</sup>''F''<sub>4</sub>(2<sup>2''n''+1</sup>) [[Ree group]]s, [[Tits group]]===
| |
| | |
| '''Simplicity:''' Simple for ''n'' ≥ 1. The derived group <sup>2</sup>''F''<sub>4</sub>(2)′ is simple of index 2
| |
| in <sup>2</sup>''F''<sub>4</sub>(2), and is called the [[Tits group]],
| |
| named for the Belgian mathematician [[Jacques Tits]].
| |
| | |
| '''Order:'''
| |
| ''q''<sup>12</sup>
| |
| (''q''<sup>6</sup> + 1)
| |
| (''q''<sup>4</sup> − 1)
| |
| (''q''<sup>3</sup> + 1)
| |
| (''q'' − 1)
| |
| where
| |
| ''q'' = 2<sup>2''n''+1</sup>.
| |
| | |
| The Tits group has order 17971200 = 2<sup>11</sup> · 3<sup>3</sup> · 5<sup>2</sup> · 13.
| |
| | |
| '''Schur multiplier:''' Trivial for ''n'' ≥ 1 and for the Tits group.
| |
| | |
| '''Outer automorphism group:'''
| |
| : 1·''f''·1
| |
| where ''f'' = 2''n'' + 1. Order 2 for the Tits group.
| |
| | |
| '''Remarks:''' Unlike the other simple groups of Lie type, the Tits group does not have a [[BN pair]], though its automorphism group does so most authors count it as a sort of honorary group of Lie type.
| |
| | |
| ===<sup>2</sup>''G''<sub>2</sub>(3<sup>2''n''+1</sup>) [[Ree group]]s===
| |
| | |
| '''Simplicity:''' Simple for ''n'' ≥ 1. The group ''<sup>2</sup>G<sub>2</sub>''(3) is not simple, but its derived group ''<sup>2</sup>G<sub>2</sub>''(3)′ is a simple subgroup of index 3.
| |
| | |
| '''Order:'''
| |
| ''q''<sup>3</sup>
| |
| (''q''<sup>3</sup> + 1)
| |
| (''q'' − 1)
| |
| where
| |
| ''q'' = 3<sup>2''n''+1</sup>
| |
| | |
| '''Schur multiplier:''' Trivial for ''n''≥1 and for <sup>2</sup>''G''<sub>2</sub>(3)′.
| |
| | |
| '''Outer automorphism group:'''
| |
| : 1·''f''·1
| |
| where ''f'' = 2''n'' + 1.
| |
| | |
| '''Other names:''' Ree(3<sup>2''n''+1</sup>), R(3<sup>2''n''+1</sup>), E<sub>2</sub><sup>*</sup>(3<sup>2''n''+1</sup>) <!-- Thompson's name, common in early 70s-->.
| |
| | |
| '''Isomorphisms:''' The derived group <sup>2</sup>''G''<sub>2</sub>(3)′ is isomorphic to ''A''<sub>1</sub>(8).
| |
| | |
| '''Remarks:''' <sup>2</sup>''G''<sub>2</sub>(3<sup>2''n''+1</sup>) has a [[doubly transitive permutation representation]] on 3<sup>3(2''n''+1)</sup> + 1 points and acts on a 7-dimensional vector space over the field with 3<sup>2''n''+1</sup> elements.
| |
| | |
| ==Sporadic groups==
| |
| === [[Mathieu group]]s ''M''<sub>11</sub>, ''M''<sub>12</sub>, ''M''<sub>22</sub>, ''M''<sub>23</sub>, ''M''<sub>24</sub> ===
| |
| {| class="wikitable"
| |
| |-
| |
| |width="15%"|
| |
| !width="17%"| [[Mathieu group M11|Mathieu group ''M''<sub>11</sub>]]
| |
| !width="17%"| [[Mathieu group M12|Mathieu group ''M''<sub>12</sub>]]
| |
| !width="17%"| [[Mathieu group M22|Mathieu group ''M''<sub>22</sub>]]
| |
| !width="17%"| [[Mathieu group M23|Mathieu group ''M''<sub>23</sub>]]
| |
| !width="17%"| [[Mathieu group M24|Mathieu group ''M''<sub>24</sub>]]
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Order:''
| |
| | 2<sup>4</sup> · 3<sup>2</sup> · 5 · 11=7920
| |
| | 2<sup>6</sup> · 3<sup>3</sup> · 5 · 11=95040
| |
| | 2<sup>7</sup> · 3<sup>2</sup> · 5 · 7 · 11 = 443520
| |
| | 2<sup>7</sup> · 3<sup>2</sup> · 5 · 7 · 11 · 23=10200960
| |
| | 2<sup>10</sup> · 3<sup>3</sup> · 5 · 7 · 11 · 23= 244823040
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Schur multiplier:''
| |
| | Trivial
| |
| | Order 2
| |
| | Cyclic of order 12{{efn| 1 = There were several mistakes made in the initial calculations of the Schur multiplier, so some older books and papers list incorrect values. (This caused an error in the title of Janko's original 1976 paper<ref>{{cite journal
| |
| | title = A new finite simple group of order 86,775,571,046,077,562,880 which possesses ''M''<sub>24</sub> and the full covering group of ''M''<sub>22</sub> as subgroups.
| |
| | journal = J. Algebra
| |
| | volume = 42
| |
| | year = 1976
| |
| | pages = 564–596
| |
| }}</ref> giving evidence for the existence of the group ''J''<sub>4</sub>. At the time it was thought that the full covering group of ''M''<sub>22</sub> was 6·''M''<sub>22</sub>. In fact ''J''<sub>4</sub> has no subgroup 12·''M''<sub>22</sub>.)}}
| |
| | Trivial
| |
| | Trivial
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Outer automorphism group:''
| |
| | Trivial
| |
| | Order 2
| |
| | Order 2
| |
| | Trivial
| |
| | Trivial
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Remarks:''
| |
| |valign=top| A 4-transitive [[permutation group]] on 11 points, and the point stabilizer in ''M''<sub>12</sub>. The subgroup fixing a point is sometimes called ''M''<sub>10</sub>, and has a subgroup of index 2 isomorphic to the alternating group A<sub>6</sub>.
| |
| |valign=top| A 5-transitive [[permutation group]] on 12 points.
| |
| |valign=top| A 3-transitive [[permutation group]] on 22 points.
| |
| |valign=top| A 4-transitive [[permutation group]] on 23 points, contained in ''M''<sub>24</sub>.
| |
| |valign=top| A 5-transitive [[permutation group]] on 24 points.
| |
| |}
| |
| | |
| === [[Janko group]]s ''J''<sub>1</sub>, ''J''<sub>2</sub>, ''J''<sub>3</sub>, ''J''<sub>4</sub> ===
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| |width="15%"|
| |
| !width="21%"| [[Janko group J1|Janko group ''J'']]<sub>1</sub>
| |
| !width="21%"| [[Janko group J2|Janko group ''J'']]<sub>2</sub>
| |
| !width="21%"| [[Janko group J3|Janko group ''J'']]<sub>3</sub>
| |
| !width="22%"| [[Janko group J4|Janko group ''J'']]<sub>4</sub>
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Order:''
| |
| | 2<sup>3</sup> · 3 · 5 · 7 · 11 · 19 = 175560
| |
| | 2<sup>7</sup> · 3<sup>3</sup> · 5<sup>2</sup> · 7 = 604800
| |
| | 2<sup>7</sup> · 3<sup>5</sup> · 5 · 17 · 19 = 50232960
| |
| | 2<sup>21</sup> · 3<sup>3</sup> · 5 · 7 · 11<sup>3</sup> · 23 · 29 · 31 · 37 · 43 = 86775571046077562880
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Schur multiplier:''
| |
| | Trivial
| |
| | Order 2
| |
| | Order 3
| |
| | Trivial
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Outer automorphism group:''
| |
| | Trivial
| |
| | Order 2
| |
| | Order 2
| |
| | Trivial
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Other names:''
| |
| | J(1), J(11)
| |
| | Hall–Janko group, HJ
| |
| | Higman–Janko–McKay group, HJM |
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Remarks:''
| |
| |valign=top| It is a subgroup of ''G''<sub>2</sub>(11), and so has a 7 dimensional representation over the field with 11 elements.
| |
| |valign=top| It is the automorphism group of a rank 3 graph on 100 points, and is also contained in ''G''<sub>2</sub>(4).
| |
| |valign=top| ''J''<sub>3</sub> seems unrelated to any other sporadic groups (or to anything else). Its triple cover has a 9 dimensional [[unitary representation]] over the field with 4 elements.
| |
| |valign=top| Has a 112 dimensional representation over the field with 2 elements.
| |
| |}
| |
| | |
| === [[Conway group]]s ''Co''<sub>1</sub>, ''Co''<sub>2</sub>, ''Co''<sub>3</sub> ===
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| |width="15%"|
| |
| !width="28%"|[[Conway group Co1|Conway group ''Co''<sub>1</sub>]]
| |
| !width="28%"|[[Conway group Co2|Conway group ''Co''<sub>2</sub>]]
| |
| !width="29%"|[[Conway group Co3|Conway group ''Co''<sub>3</sub>]]
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Order:''
| |
| | 2<sup>21</sup> · 3<sup>9</sup> · 5<sup>4</sup> · 7<sup>2</sup> · 11 · 13 · 23 = 4157776806543360000
| |
| | 2<sup>18</sup> · 3<sup>6</sup> · 5<sup>3</sup> · 7 · 11 · 23 = 42305421312000
| |
| | 2<sup>10</sup> · 3<sup>7</sup> · 5<sup>3</sup> · 7 · 11 · 23 = 495766656000
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Schur multiplier:''
| |
| | Order 2
| |
| | Trivial
| |
| | Trivial
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Outer automorphism group:''
| |
| | Trivial
| |
| | Trivial
| |
| | Trivial
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Other names:''
| |
| |·1
| |
| |·2
| |
| |·3, C<sub>3</sub>
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Remarks:''
| |
| |valign=top| The perfect double cover of ''Co''<sub>1</sub> is the automorphism group of the [[Leech lattice]], and is sometimes denoted by ·0.
| |
| |valign=top| Subgroup of ''Co''<sub>1</sub>; fixes a norm 4 vector in the [[Leech lattice]].
| |
| |valign=top| Subgroup of ''Co''<sub>1</sub>; fixes a norm 6 vector in the [[Leech lattice]]. It has a doubly transitive permutation representation on 276 points.
| |
| |}
| |
| | |
| === [[Fischer group]]s ''Fi''<sub>22</sub>, ''Fi''<sub>23</sub>, ''Fi''<sub>24</sub>' ===
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| |width="15%"|
| |
| !width="28%"| [[Fischer group Fi22| Fischer group ''Fi''<sub>22</sub>]]
| |
| !width="28%"| [[Fischer group Fi23| Fischer group ''Fi''<sub>23</sub>]]
| |
| !width="29%"| [[Fischer group Fi24| Fischer group ''Fi''<sub>24</sub>']]
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Order:''
| |
| | 2<sup>17</sup> · 3<sup>9</sup> · 5<sup>2</sup> · 7 · 11 · 13 = 64561751654400
| |
| | 2<sup>18</sup> · 3<sup>13</sup> · 5<sup>2</sup> · 7 · 11 · 13 · 17 · 23 = 4089470473293004800
| |
| | 2<sup>21</sup> · 3<sup>16</sup> · 5<sup>2</sup> · 7<sup>3</sup> · 11 · 13 · 17 · 23 · 29 = 1255205709190661721292800
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Schur multiplier:''
| |
| | Order 6
| |
| | Trivial
| |
| | Order 3
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Outer automorphism group:''
| |
| | Order 2
| |
| | Trivial
| |
| | Order 2
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Other names:''
| |
| | ''M''(22)
| |
| | ''M''(23)
| |
| | ''M''(24)′, ''F''<sub>3+</sub>
| |
| |-
| |
| !valign=top style="text-align: right;"| ''Remarks:''
| |
| |valign=top| A 3-transposition group whose double cover is contained in ''Fi''<sub>23</sub>.
| |
| |valign=top| A 3-transposition group contained in ''Fi''<sub>24</sub>.
| |
| |valign=top| The triple cover is contained in the monster group.
| |
| |}
| |
| | |
| ===[[Higman–Sims group]] ''HS''===
| |
| '''Order:''' 2<sup>9</sup> · 3<sup>2</sup> · 5<sup>3</sup>· 7 · 11 = 44352000
| |
| | |
| '''Schur multiplier:''' Order 2.
| |
| | |
| '''Outer automorphism group:''' Order 2.
| |
| | |
| '''Remarks:''' It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in ''Co''<sub>3</sub>.
| |
| | |
| ===[[McLaughlin group (mathematics)|McLaughlin group]] ''McL''===
| |
| '''Order:''' 2<sup>7</sup> · 3<sup>6</sup> · 5<sup>3</sup>· 7 · 11 = 898128000
| |
| | |
| '''Schur multiplier:''' Order 3.
| |
| | |
| '''Outer automorphism group:''' Order 2.
| |
| | |
| '''Remarks:''' Acts as a rank 3 permutation group on the McLaughlin graph with 275 points, and is contained in ''Co''<sub>3</sub>.
| |
| | |
| ===[[Held group]] ''He''===
| |
| '''Order:'''
| |
| 2<sup>10</sup> · 3<sup>3</sup> · 5<sup>2</sup>· 7<sup>3</sup>· 17 = 4030387200
| |
| | |
| '''Schur multiplier:''' Trivial.
| |
| | |
| '''Outer automorphism group:''' Order 2.
| |
| | |
| '''Other names:''' Held–Higman–McKay group, HHM, ''F''<sub>7</sub>, HTH
| |
| | |
| '''Remarks:''' Centralizes an element of order 7 in the monster group.
| |
| | |
| ===[[Rudvalis group]] ''Ru''===
| |
| '''Order:'''
| |
| 2<sup>14</sup> · 3<sup>3</sup> · 5<sup>3</sup>· 7 · 13 · 29 = 145926144000
| |
| | |
| '''Schur multiplier:''' Order 2.
| |
| | |
| '''Outer automorphism group:''' Trivial.
| |
| | |
| '''Remarks:''' The double cover acts on a 28 dimensional lattice over the [[Gaussian integer]]s.
| |
| | |
| ===[[Suzuki sporadic group]] ''Suz''===
| |
| '''Order:''' 2<sup>13</sup> · 3<sup>7</sup> · 5<sup>2</sup>· 7 · 11 · 13 = 448345497600
| |
| | |
| '''Schur multiplier:''' Order 6.
| |
| | |
| '''Outer automorphism group:''' Order 2.
| |
| | |
| '''Other names:''' ''Sz''
| |
| | |
| '''Remarks:''' The 6 fold cover acts on a 12 dimensional lattice over the [[Eisenstein integer]]s. It is not related to the Suzuki groups of Lie type.
| |
| | |
| ===[[O'Nan group]] ''O'N''===
| |
| '''Order:'''
| |
| 2<sup>9</sup> · 3<sup>4</sup> · 5 · 7<sup>3</sup> · 11 · 19 · 31 = 460815505920
| |
| | |
| '''Schur multiplier:''' Order 3.
| |
| | |
| '''Outer automorphism group:''' Order 2.
| |
| | |
| '''Other names:''' O'Nan–Sims group, O'NS, O–S
| |
| | |
| '''Remarks:'''
| |
| The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.
| |
| | |
| ===[[Harada–Norton group]] ''HN''===
| |
| '''Order:'''
| |
| 2<sup>14</sup> · 3<sup>6</sup> · 5<sup>6</sup> · 7 · 11 · 19 = 273030912000000
| |
| | |
| '''Schur multiplier:''' Trivial.
| |
| | |
| '''Outer automorphism group:''' Order 2.
| |
| | |
| '''Other names:''' ''F''<sub>5</sub>, ''D''
| |
| | |
| '''Remarks:''' Centralizes an element of order 5 in the monster group.
| |
| | |
| ===[[Lyons group]] ''Ly''===
| |
| '''Order:'''
| |
| 2<sup>8</sup> · 3<sup>7</sup> · 5<sup>6</sup> · 7 · 11 · 31 · 37 · 67 = 51765179004000000
| |
| | |
| '''Schur multiplier:''' Trivial.
| |
| | |
| '''Outer automorphism group:''' Trivial.
| |
| | |
| '''Other names:''' Lyons–Sims group, ''LyS''
| |
| | |
| '''Remarks:''' Has a 111 dimensional representation over the field with 5 elements.
| |
| | |
| ===[[Thompson group (finite)|Thompson group]] ''Th''===
| |
| '''Order:''' 2<sup>15</sup> · 3<sup>10</sup> · 5<sup>3</sup> · 7<sup>2</sup> · 13 · 19 · 31 = 90745943887872000
| |
| | |
| '''Schur multiplier:''' Trivial.
| |
| | |
| '''Outer automorphism group:''' Trivial.
| |
| | |
| '''Other names:''' ''F''<sub>3</sub>, ''E''
| |
| | |
| '''Remarks:''' Centralizes an element of order 3 in the monster, and is contained in ''E''<sub>8</sub>(3), so has a 248-dimensional representation over the field with 3 elements.
| |
| | |
| ===[[Baby Monster group]] ''B''===
| |
| '''Order:'''
| |
| : 2<sup>41</sup> · 3<sup>13</sup> · 5<sup>6</sup> · 7<sup>2</sup> · 11 · 13 · 17 · 19 · 23 · 31 · 47
| |
| : = 4154781481226426191177580544000000
| |
| | |
| '''Schur multiplier:''' Order 2.
| |
| | |
| '''Outer automorphism group:''' Trivial.
| |
| | |
| '''Other names:''' ''F''<sub>2</sub>
| |
| | |
| '''Remarks:''' The double cover is contained in the monster group. It has a representation of dimension 4371 over the complex numbers (with no nontrivial invariant product), and a representation of dimension 4370 over the field with 2 elements preserving a commutative but non-associative product.
| |
| | |
| ===Fischer–Griess [[Monster group]] ''M''===
| |
| '''Order:'''
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| : 2<sup>46</sup> · 3<sup>20</sup> · 5<sup>9</sup> · 7<sup>6</sup> · 11<sup>2</sup> · 13<sup>3</sup> · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
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| : = 808017424794512875886459904961710757005754368000000000
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| '''Schur multiplier:''' Trivial.
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| '''Outer automorphism group:''' Trivial.
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| '''Other names:''' ''F''<sub>1</sub>, ''M''<sub>1</sub>, Monster group, Friendly giant, Fischer's monster.
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| '''Remarks:''' Contains all but 6 of the other sporadic groups as subquotients. Related to [[monstrous moonshine]]. The monster is the automorphism group of the 196884 dimensional [[Griess algebra]] and the infinite dimensional monster [[vertex operator algebra]], and acts naturally on the [[monster Lie algebra]].
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| ==Non-cyclic simple groups of small order==
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| | |
| {| class="wikitable"
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| |-
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| ! [[Order (group theory)|Order]]!![[Factorization|Factored order]]!!Group!![[Schur multiplier]]!![[Outer automorphism group]]
| |
| |-
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| | 60||2<sup>2</sup> · 3 · 5||A<sub>5</sub> = ''A''<sub>1</sub>(4) = ''A''<sub>1</sub>(5)||2||2
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| |-
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| | 168||2<sup>3</sup> · 3 · 7||''A''<sub>1</sub>(7) = ''A''<sub>2</sub>(2)||2||2
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| |-
| |
| | 360||2<sup>3</sup> · 3<sup>2</sup> · 5||A<sub>6</sub> = ''A''<sub>1</sub>(9) = ''B''<sub>2</sub>(2)′||6||2×2
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| |-
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| | 504||2<sup>3</sup> · 3<sup>2</sup> · 7||''A''<sub>1</sub>(8) = <sup>2</sup>''G''<sub>2</sub>(3)′||1||3
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| |-
| |
| | 660||2<sup>2</sup> · 3 · 5 · 11||''A''<sub>1</sub>(11)||2||2
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| |-
| |
| | 1092||2<sup>2</sup> · 3 · 7 · 13||''A''<sub>1</sub>(13)||2||2
| |
| |-
| |
| | 2448||2<sup>4</sup> · 3<sup>2</sup> · 17||''A''<sub>1</sub>(17)||2||2
| |
| |-
| |
| | 2520||2<sup>3</sup> · 3<sup>2</sup> · 5 · 7||A<sub>7</sub>||6||2
| |
| |-
| |
| | 3420||2<sup>2</sup> · 3<sup>2</sup> · 5 · 19||''A''<sub>1</sub>(19)||2||2
| |
| |-
| |
| | 4080||2<sup>4</sup> · 3 · 5 · 17||''A''<sub>1</sub>(16)||1||4
| |
| |-
| |
| | 5616||2<sup>4</sup> · 3<sup>3</sup> · 13||''A''<sub>2</sub>(3)||1||2
| |
| |-
| |
| | 6048||2<sup>5</sup> · 3<sup>3</sup> · 7||<sup>2</sup>''A''<sub>2</sub>(9) = ''G''<sub>2</sub>(2)′||1||2
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| |-
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| | 6072||2<sup>3</sup> · 3 · 11 · 23||''A''<sub>1</sub>(23)||2||2
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| |-
| |
| | 7800||2<sup>3</sup> · 3 · 5<sup>2</sup> · 13||''A''<sub>1</sub>(25)||2||2×2
| |
| |-
| |
| | 7920||2<sup>4</sup> · 3<sup>2</sup> · 5 · 11||''M''<sub>11</sub>||1||1
| |
| |-
| |
| | 9828||2<sup>2</sup> · 3<sup>3</sup> · 7 · 13||''A''<sub>1</sub>(27)||2||6
| |
| |-
| |
| | 12180||2<sup>2</sup> · 3 · 5 · 7 · 29||''A''<sub>1</sub>(29)||2||2
| |
| |-
| |
| | 14880||2<sup>5</sup> · 3 · 5 · 31||''A''<sub>1</sub>(31)||2||2
| |
| |-
| |
| | 20160||2<sup>6</sup> · 3<sup>2</sup> · 5 · 7||''A''<sub>3</sub>(2) = A<sub>8</sub>||2||2
| |
| |-
| |
| | 20160||2<sup>6</sup> · 3<sup>2</sup> · 5 · 7||''A''<sub>2</sub>(4) ||3×4<sup>2</sup>||D<sub>12</sub>
| |
| |-
| |
| | 25308||2<sup>2</sup> · 3<sup>2</sup> · 19 · 37||''A''<sub>1</sub>(37)||2||2
| |
| |-
| |
| | 25920||2<sup>6</sup> · 3<sup>4</sup> · 5 ||<sup>2</sup>''A''<sub>3</sub>(4) = ''B''<sub>2</sub>(3)||2||2
| |
| |-
| |
| | 29120||2<sup>6</sup> · 5 · 7 · 13 ||<sup>2</sup>''B''<sub>2</sub>(8) ||2<sup>2</sup>||3
| |
| |-
| |
| | 32736||2<sup>5</sup> · 3 · 11 · 31||''A''<sub>1</sub>(32)||1||5
| |
| |-
| |
| | 34440||2<sup>3</sup> · 3 · 5 · 7 · 41 ||''A''<sub>1</sub>(41)||2||2
| |
| |-
| |
| | 39732||2<sup>2</sup> · 3 · 7 · 11 · 43 ||''A''<sub>1</sub>(43)||2||2
| |
| |-
| |
| | 51888||2<sup>4</sup> · 3 · 23 · 47||''A''<sub>1</sub>(47)||2||2
| |
| |-
| |
| | 58800||2<sup>4</sup> · 3 · 5<sup>2</sup> · 7<sup>2</sup>||''A''<sub>1</sub>(49)||2||2<sup>2</sup>
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| |-
| |
| | 62400||2<sup>6</sup> · 3 · 5<sup>2</sup> · 13||<sup>2</sup>''A''<sub>2</sub>(16)||1||4
| |
| |-
| |
| | 74412||2<sup>2</sup> · 3<sup>3</sup> · 13 · 53||''A''<sub>1</sub>(53)||2||2
| |
| |-
| |
| | 95040||2<sup>6</sup> · 3<sup>3</sup> · 5 · 11||''M''<sub>12</sub>||2||2
| |
| |}
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| (Complete for orders less than 100,000)
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| {{harvtxt|Hall|1972}} lists the 56 non-cyclic simple groups of order less than a million.
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| ==See also==
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| *[[List of small groups]]
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| ==Notes==
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| {{notelist}}
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| ==References==
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| {{Reflist}}
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| ==Further reading==
| |
| | |
| *''Simple Groups of Lie Type'' by [[Roger Carter (mathematician)|Roger W. Carter]], ISBN 0-471-50683-4
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| * [[John Horton Conway|Conway, J. H]].; Curtis, R. T.; [[Simon P. Norton|Norton, S. P.]]; Parker, R. A.; and [[Robert Arnott Wilson|Wilson, R. A.]]: "''Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups.''" Oxford, England 1985.
| |
| * [[Daniel Gorenstein]], Richard Lyons, Ronald Solomon ''The Classification of the Finite Simple Groups'' [http://www.ams.org/online_bks/surv401/ (volume 1)], AMS, 1994 [http://www.ams.org/online_bks/surv402/ (volume 2)], AMS,
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| *{{Citation | last1=Hall | first1=Marshall Jr. | title=Simple groups of order less than one million | doi=10.1016/0021-8693(72)90090-7 | id={{MR|0285603}} | year=1972 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=20 | pages=98–102}}
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| *{{Citation | last1=Wilson | first1=Robert A. | authorlink = Robert Arnott Wilson | title=The finite simple groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | year=2009 | zbl=05622792 | volume=251 }}
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| * [http://brauer.maths.qmul.ac.uk/Atlas/v3/ Atlas of Finite Group Representations]: contains [[Group representation|representations]] and other data for many finite simple groups, including the sporadic groups.
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| ==External links==
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| * [http://www.eleves.ens.fr:8080/home/madore/math/simplegroups.html Orders of non abelian simple groups] up to order 10,000,000,000.
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| [[Category:Mathematics-related lists|Finite simple groups]]
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| [[Category:Group theory]]
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| [[Category:Sporadic groups]]
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