Abstract group 212427600.a: $\SL(3,11)$

• Label: 212427600.a
• Order: \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{3} \cdot 19 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{3} \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $133$
• Trans deg.: $133$
• Rank: $2$

Group information

Description:	$\SL(3,11)$
Order:	\(212427600\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{3} \cdot 19 \)
Exponent:	\(175560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \)
Automorphism group:	$\PSL(3,11).C_2$, of order \(424855200\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{3} \cdot 19 \)
Composition factors:	$\SL(3,11)$
Derived length:	$0$

This group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).

Group statistics

Order	1	2	3	4	5	6	7	8	10	11	12	15	19	20	22	24	30	40	55	60	110	120	133
Elements	1	16093	1770230	1770230	4312924	1770230	3194400	3540460	21307132	1771560	3540460	7080920	9583200	7080920	1931160	7080920	7080920	14161840	7724640	14161840	7724640	28323680	57499200	212427600
Conjugacy classes	1	1	1	1	6	1	2	2	14	2	2	4	6	4	1	4	4	8	4	8	4	16	36	132
Divisions	1	1	1	1	2	1	1	1	4	2	1	1	1	1	1	1	1	1	1	1	1	1	1	28
Autjugacy classes	1	1	1	1	4	1	1	1	8	2	2	2	3	2	1	2	2	4	2	4	2	8	18	73

Minimal presentations

Permutation degree:	$133$
Transitive degree:	$133$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	132	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\SL(3,11)$, $\PSL(3,11)$, $\PGL(3,11)$, $\PGammaL(3,11)$, $\PSigmaL(3,11)$
Permutation group:	Degree $133$ $\langle(1,2,5,12,10,4,9,22,51,93,94,125,80,38,79,70,32,47,28,60,36,77,103,131,100,132,98,54,97,83,41,18,44,61,106,114,66,30,64,109,122,74,34,68,31,13,29,62,108,95,52,63,53,23,43,85,90,121,99,133,119,69,118,107,76,112,65,73,117,105,113,110,82,39,16,6,14,33,71,88,102,81,40,78,124,116,130,92,56,24,55,48,20,8,19,46,89,127,120,72,96,91,49,21,50,26,11,25,58,87,45,67,115,129,126,84,42,17,7,3) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\SL(3,11)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 2 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $\SL(3,11)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $\He_{11}$
19-Sylow subgroup:	$P_{ 19 } \simeq$ $C_{19}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 880 (order at least 241395) are shown, as well as all normal subgroups of any index.

Order 212427600: $\SL(3,11)$

Order 1597200: $C_{11}^2.\GL(2,11)$ x 2

Order 798600: $C_{11}^2.C_{10}.\PSL(2,11)$ x 2

Order 319440: $C_{11}:D_{11}.\PSL(2,11).C_2$ x 2

Order 1331: $\He_{11}$

Order 1320: $\PGL(2,11)$

Order 600: $C_5^2:S_4$

Order 399: $C_{133}:C_3$

Order 168: $\PSL(2,7)$

Order 25: $C_5^2$

Order 19: $C_{19}$

Order 16: $\SD_{16}$

Order 7: $C_7$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 212427600: $\SL(3,11)$ ($C_1$)

Order 1: $C_1$ ($\SL(3,11)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

The $132 \times 132$ character table is not available for this group.

Rational character table

The $28 \times 28$ rational character table is not available for this group.