Abstract group $\PSL(3,4):C_2$

• Label: 40320.o
• Order: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $42$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$\PSL(3,4):C_2$
Order:	\(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	$\PSL(3,4):D_6$, of order \(241920\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) (generators)
Outer automorphisms:	$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Composition factors:	$C_2$, $\PSL(3,4)$
Derived length:	$1$

This group is nonabelian, almost simple, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	7	8
Elements	1	595	2240	6300	8064	2240	5760	15120	40320
Conjugacy classes	1	2	1	4	1	1	1	3	14
Divisions	1	2	1	4	1	1	1	3	14
Autjugacy classes	1	2	1	2	1	1	1	1	10

Dimension	1	20	35	64	90	126
Irr. complex chars.	2	2	6	2	1	1	14
Irr. rational chars.	2	2	6	2	1	1	14

Minimal Presentations

Permutation degree:	$42$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	$4623$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	20	20
Arbitrary	20	20	20

Constructions

Permutation group:	Degree $42$ $\langle(1,9,5,12,3)(2,8,19,17,4)(6,18,21,16,10)(7,11,13,20,14)(22,38,42,31,25) \!\cdots\! \rangle$
Transitive group:	42T932				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\PSL(3,4)$ $\,\rtimes\,$ $C_2$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 42.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 100708 subgroups in 151 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\PSL(3,4):C_2$
Commutator:	$G' \simeq$ $\PSL(3,4)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\PSL(3,4):C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\PSL(3,4):C_2$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\PSL(3,4):C_2$
Socle:	$\operatorname{soc} \simeq$ $\PSL(3,4)$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 40320: $\PSL(3,4):C_2$

Order 20160: $\PSL(3,4)$

Order 960: $C_2^4:A_5$

Order 720: $A_6.C_2$ x 3

Order 384: $C_4^2.S_4$

Order 360: $A_6$ x 3

Order 336: $\PGL(2,7)$ x 3

Order 192: $C_4^2:A_4$

Order 168: $\GL(3,2)$ x 3

Order 160: $C_2^4:D_5$

Order 144: $C_2\times \PSU(3,2)$

Order 128: $C_4^2.D_4$

Order 120: $S_5$

Order 96: $C_4^2:S_3$ x 3, $C_2^2:S_4$

Order 80: $C_2^4:C_5$

Order 72: $\PSU(3,2)$ x 4, $C_2\times C_3^2:C_4$ x 3

Order 64: $C_4^2.C_4$ x 3, $D_4.D_4$ x 3, $C_4^2:C_2^2$

Order 60: $A_5$ x 4

Order 48: $C_4^2:C_3$ x 3, $C_2^2:A_4$

Order 42: $F_7$

Order 36: $C_3^2:C_4$ x 6, $C_6:S_3$

Order 32: $C_4.D_4$ x 3, $D_8:C_2$ x 3, $C_4^2:C_2$ x 3, $C_4\wr C_2$ x 3, $C_4.C_2^3$, $C_2^2\wr C_2$

Order 24: $S_4$ x 4

Order 21: $C_7:C_3$

Order 20: $F_5$

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $D_4:C_2$ x 4, $\SD_{16}$ x 3, $D_8$ x 3, $\OD_{16}$ x 3, $C_2^2:C_4$ x 3, $C_4^2$ x 3, $C_2\times D_4$ x 3, $C_2\times Q_8$ x 2, $C_2^4$

Order 14: $D_7$

Order 12: $A_4$ x 4, $D_6$

Order 10: $D_5$

Order 9: $C_3^2$

Order 8: $D_4$ x 7, $C_2\times C_4$ x 6, $Q_8$ x 4, $C_8$ x 3, $C_2^3$

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 40320: $\PSL(3,4):C_2$

Order 20160: $\PSL(3,4)$

Order 960: $C_2^4:A_5$

Order 720: $A_6.C_2$

Order 384: $C_4^2.S_4$

Order 360: $A_6$

Order 336: $\PGL(2,7)$

Order 192: $C_4^2:A_4$

Order 168: $\GL(3,2)$

Order 160: $C_2^4:D_5$

Order 144: $C_2\times \PSU(3,2)$

Order 128: $C_4^2.D_4$

Order 120: $S_5$

Order 96: $C_4^2:S_3$, $C_2^2:S_4$

Order 80: $C_2^4:C_5$

Order 72: $\PSU(3,2)$ x 2, $C_2\times C_3^2:C_4$

Order 64: $C_4^2.C_4$, $C_4^2:C_2^2$, $D_4.D_4$

Order 60: $A_5$ x 2

Order 48: $C_2^2:A_4$, $C_4^2:C_3$

Order 42: $F_7$

Order 36: $C_3^2:C_4$ x 2, $C_6:S_3$

Order 32: $C_4.D_4$, $C_4.C_2^3$, $D_8:C_2$, $C_4^2:C_2$, $C_2^2\wr C_2$, $C_4\wr C_2$

Order 24: $S_4$ x 2

Order 21: $C_7:C_3$

Order 20: $F_5$

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $D_4:C_2$ x 2, $C_2\times Q_8$ x 2, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_2^2:C_4$, $C_4^2$, $C_2^4$, $C_2\times D_4$

Order 14: $D_7$

Order 12: $A_4$ x 2, $D_6$

Order 10: $D_5$

Order 9: $C_3^2$

Order 8: $D_4$ x 3, $Q_8$ x 2, $C_2\times C_4$ x 2, $C_2^3$, $C_8$

Order 7: $C_7$

Order 6: $S_3$ x 2, $C_6$

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 40320: $\PSL(3,4):C_2$ ($C_1$)

Order 20160: $\PSL(3,4)$ ($C_2$)

Order 1: $C_1$ ($\PSL(3,4):C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 40320: $\PSL(3,4):C_2$ ($C_1$)

Order 20160: $\PSL(3,4)$ ($C_2$)

Order 1: $C_1$ ($\PSL(3,4):C_2$)

Series

Derived series	$\PSL(3,4):C_2$	$\rhd$	$\PSL(3,4)$
Chief series	$\PSL(3,4):C_2$	$\rhd$	$\PSL(3,4)$	$\rhd$	$C_1$
Lower central series	$\PSL(3,4):C_2$	$\rhd$	$\PSL(3,4)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

		1A	2A	2B	3A	4A	4B	4C	4D	5A	6A	7A	8A	8B	8C
	Size	1	280	315	2240	1260	1260	1260	2520	8064	2240	5760	5040	5040	5040
	2 P	1A	1A	1A	3A	2B	2B	2B	2B	5A	3A	7A	4B	4C	4A
	3 P	1A	2A	2B	1A	4A	4B	4C	4D	5A	2A	7A	8B	8C	8A
	5 P	1A	2A	2B	3A	4A	4B	4C	4D	1A	6A	7A	8B	8C	8A
	7 P	1A	2A	2B	3A	4A	4B	4C	4D	5A	6A	1A	8B	8C	8A
40320.o.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
40320.o.1b		$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$
40320.o.20a		$20$	$-2$	$4$	$2$	$0$	$0$	$0$	$2$	$0$	$-2$	$-1$	$0$	$0$	$0$
40320.o.20b		$20$	$2$	$4$	$2$	$0$	$0$	$0$	$-2$	$0$	$2$	$-1$	$0$	$0$	$0$
40320.o.35a		$35$	$1$	$3$	$-1$	$-1$	$-1$	$3$	$1$	$0$	$1$	$0$	$-1$	$1$	$-1$
40320.o.35b		$35$	$1$	$3$	$-1$	$-1$	$3$	$-1$	$1$	$0$	$1$	$0$	$1$	$-1$	$-1$
40320.o.35c		$35$	$1$	$3$	$-1$	$3$	$-1$	$-1$	$1$	$0$	$1$	$0$	$-1$	$-1$	$1$
40320.o.35d		$35$	$-1$	$3$	$-1$	$-1$	$-1$	$3$	$-1$	$0$	$-1$	$0$	$1$	$-1$	$1$
40320.o.35e		$35$	$-1$	$3$	$-1$	$-1$	$3$	$-1$	$-1$	$0$	$-1$	$0$	$-1$	$1$	$1$
40320.o.35f		$35$	$-1$	$3$	$-1$	$3$	$-1$	$-1$	$-1$	$0$	$-1$	$0$	$1$	$1$	$-1$
40320.o.64a		$64$	$8$	$0$	$1$	$0$	$0$	$0$	$0$	$-1$	$-1$	$1$	$0$	$0$	$0$
40320.o.64b		$64$	$-8$	$0$	$1$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$	$0$	$0$	$0$
40320.o.90a		$90$	$0$	$-6$	$0$	$2$	$2$	$2$	$0$	$0$	$0$	$-1$	$0$	$0$	$0$
40320.o.126a		$126$	$0$	$-2$	$0$	$-2$	$-2$	$-2$	$0$	$1$	$0$	$0$	$0$	$0$	$0$