Abstract group 1931325984.a: $\ASL(3,7)$

• Label: 1931325984.a
• Order: \( 2^{5} \cdot 3^{3} \cdot 7^{6} \cdot 19 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \cdot 7^{6} \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $343$
• Trans deg.: not computed
• Rank: $2$

Group information

Description:	$\ASL(3,7)$
Order:	\(1931325984\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7^{6} \cdot 19 \)
Exponent:	\(6384\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Automorphism group:	Group of order \(11587955904\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 7^{6} \cdot 19 \)
Composition factors:	$C_3$, $C_7$ x 3, $\PSL(3,7)$
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable). Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	19	21	24	28	42	48	56	57
Elements	1	136857	7664678	5747994	116875878	40353606	11495988	80471916	46805094	160943832	203297472	126690480	160943832	34487964	137951856	321887664	68975928	406594944	1931325984
Conjugacy classes	1	1	3	1	5	14	2	2	4	4	6	9	4	1	3	8	2	12	82
Divisions	1	1	2	1	3	14	1	1	4	1	1	5	1	1	2	1	1	1	42
Autjugacy classes	1	1	3	1	5	6	2	2	3	4	6	5	4	1	3	8	2	12	69

Dimension	1	56	57	96	114	152	192	288	342	343	399	456	684	798	912	1026	1368	1728	2052	2394	2736	3456	4104	16416
Irr. complex chars.	1	1	5	6	0	3	0	18	22	1	5	3	0	0	0	2	2	0	3	1	2	0	0	7	82
Irr. rational chars.	1	1	1	0	2	3	3	0	2	1	1	1	2	2	1	2	4	1	1	1	3	1	1	7	42

Minimal presentations

Permutation degree:	$343$
Transitive degree:	not computed
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	342	342	342
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\ASL(3,7)$, $\ASigmaL(3,7)$
Permutation group:	Degree $343$ $\langle(1,2,6,4,9,7,3)(5,16,49,17,43,14,18)(8,23,66,96,109,71,24)(10,29,27,80,30,84,31) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\ASL(3,7)$.

Homology

Abelianization:	$C_1 $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

Subgroup data has not been computed.

Character theory

Complex character table

See the $82 \times 82$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $42 \times 42$ rational character table.