Abstract group 193536.b: $C_2^5.(S_3^2\times \GL(3,2))$

• Label: 193536.b
• Order: \( 2^{10} \cdot 3^{3} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_2^5.(S_3^2\times \GL(3,2))$
Order:	\(193536\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$(S_3\times S_4).C_2^4.\PSL(2,7)$, of order \(387072\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 7, $C_3$ x 2, $\PSL(2,7)$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	12	14	21	28	42	84
Elements	1	3679	6074	29088	48566	384	50832	14976	9984	9216	16128	4608	193536
Conjugacy classes	1	23	7	32	41	2	31	10	6	4	6	2	165
Divisions	1	23	7	32	41	1	31	5	3	2	3	1	150
Autjugacy classes	1	17	7	22	35	2	23	10	6	4	6	2	135

Dimension	1	2	3	4	6	7	8	9	12	14	16	18	21	24	28	32	36	42	48	56	63	84	126
Irr. complex chars.	4	4	12	1	14	12	4	8	6	16	4	8	20	5	7	1	2	18	2	1	8	4	4	165
Irr. rational chars.	4	4	4	1	10	12	4	0	8	16	4	8	20	6	7	1	4	18	2	1	8	4	4	150

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	42	42	42
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(1,2)(4,6)(8,14)(9,11)(10,12)(13,15), (1,2), (4,7,6,5)(8,11,9,14)(10,15,13,12) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^5:\GL(3,2))$ . $S_3^2$	$C_2^5$ . $(S_3^2\times \GL(3,2))$	$(C_2^3:D_6^2)$ . $\PSL(2,7)$	$(S_4\times C_2^3:\GL(3,2))$ . $S_3$	all 37
Aut. group:	$\Aut(C_2^2\times C_6\times \SL(2,3))$	$\Aut(C_2^2\times A_4\times D_6)$

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 39 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 $C_2^5:(C_3^2\times \GL(3,2))$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 108 (order at least 1792) are shown, as well as all normal subgroups of any index.

Order 193536: $C_2^5.(S_3^2\times \GL(3,2))$

Order 96768: $S_3\times A_4\times C_2^3:\GL(3,2)$, $(C_2^2\times C_6):(S_4\times \GL(3,2))$, $C_3\times S_4\times C_2^3:\GL(3,2)$

Order 64512: $C_2^5.(D_6\times \GL(3,2))$, $(C_2^4\times S_4).\GL(3,2)$

Order 48384: $S_3^2\times C_2^3:\GL(3,2)$, $C_2^5:(C_3^2\times \GL(3,2))$

Order 32256: $S_4\times C_2^3:\GL(3,2)$ x 3, $(D_6\times C_2^4).\GL(3,2)$ x 2, $C_2^5:(S_3\times \GL(3,2))$ x 2, $A_4\times C_2^4:\GL(3,2)$, $D_{12}\times C_2^3:\GL(3,2)$, $C_2^4.(D_6\times \GL(3,2))$, $C_3\times D_4\times C_2^3.\PSL(2,7)$

Order 27648: $(C_2^2\times D_6).S_4^2$, $(C_2^2\times D_6).S_4^2$

Order 24192: $C_6^2:C_2^2:\GL(3,2)$ x 2, $S_3\times S_4\times \GL(3,2)$ x 2, $F_8:C_6^2:D_6$, $C_6^2:C_2^2:\GL(3,2)$

Order 21504: $(D_4\times C_2^4).\GL(3,2)$

Order 16128: $S_3\times C_2^4:\GL(3,2)$ x 6, $C_6\times C_2^4:\GL(3,2)$ x 2, $A_4\times C_2^3:\GL(3,2)$ x 2, $C_2^3:\GL(3,2)\times C_{12}$, $C_2^4.(S_3\times \GL(3,2))$

Order 13824: $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^5:(A_4\times S_3^2)$, $C_3\times C_2^3:S_4^2$, $Q_8:A_4^2:D_6$, $Q_8:A_4^2:D_6$, $C_2^3.A_4^2:D_6$, $C_2^6:(C_6\times S_3^2)$, $(C_2^2\times C_6):S_4^2$, $(C_2^2\times C_6):S_4^2$, $C_2^5.(A_4\times S_3^2)$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6):S_4^2$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$

Order 12096: $C_3:S_4\times \GL(3,2)$ x 2, $C_3\times S_4\times \GL(3,2)$ x 2, $S_3\times A_4\times \GL(3,2)$ x 2, $F_8:C_6^2:C_6$, $F_8:C_3^2:S_4$, $F_8:C_3^2\times S_4$, $C_3^2\times C_2^3:\GL(3,2)$

Order 10752: $D_4\times C_2^3:\GL(3,2)$ x 4, $C_2^6.\GL(3,2)$ x 2, $C_4\times C_2^4:\GL(3,2)$

Order 9216: $C_2^5.(D_6\times S_4)$, $C_2^6.D_6^2$, $C_2^4.S_4^2$, $C_2^6.D_6^2$, $C_2^4.S_4^2$

Order 8064: $S_3\times C_2^3:\GL(3,2)$ x 6, $C_6\times C_2^3:\GL(3,2)$ x 3, $S_3\times D_4\times \GL(3,2)$ x 2, $C_2\times S_4\times \GL(3,2)$ x 2, $S_3\times S_4\times F_8$, $(D_{12}\times F_8):C_6$, $F_8:C_6\times S_4$

Order 6912: $D_6\times S_4^2$ x 3, $A_4\times C_2^4:S_3^2$ x 2, $(A_4\times C_2^4):S_3^2$ x 2, $A_4\times C_2^4:S_3^2$ x 2, $(A_4\times C_2^4):S_3^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $C_3\times C_2^2:S_4^2$ x 2, $C_3\times C_2^2.S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $C_2^6:(C_3\times S_3^2)$, $C_2^2:A_4^2:D_6$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:S_3$, $Q_8:A_4^2:S_3$, $C_2^5:(C_3^2\times S_4)$, $S_3\times C_2^3:A_4^2$, $Q_8:A_4^2:C_6$, $C_2^6:(C_3\times S_3^2)$, $(A_4\times C_2^4):C_6^2$, $S_3\times Q_8:A_4^2$, $Q_8:A_4^2:S_3$, $C_2^6:(C_3\times S_3^2)$, $C_2^4.(A_4\times S_3^2)$, $C_2^3:S_4\times S_3^2$, $C_2^5:(C_3^2\times S_4)$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $C_2^3:S_4\times S_3^2$

Order 6048: $S_3^2\times \GL(3,2)$ x 2, $C_3\times A_4\times \GL(3,2)$ x 2, $A_4\times F_8:C_3^2$, $F_8:C_3\times S_3^2$

Order 5376: $C_2^5:\GL(3,2)$ x 7, $C_4\times C_2^3:\GL(3,2)$ x 2

Order 4608: $C_2^3:S_4^2$ x 10, $C_2^3:S_4^2$ x 6, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5:(S_3\times S_4)$ x 2, $(C_2^2\times D_6).\GL(2,\mathbb{Z}/4)$ x 2, $D_6\times C_2^4:S_4$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^4.(D_6\times S_4)$ x 2, $C_2^3:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $(C_6\times A_4).C_2^6$, $C_{12}:C_2^4:S_4$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5:(C_6\times S_4)$, $(C_6\times \GL(2,\mathbb{Z}/4)):D_4$, $C_2^3:S_4\times D_{12}$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.(A_4\times D_6)$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2\wr C_6.D_6$, $C_2.C_2^6.C_3.D_6$, $A_4\times C_2^4:S_4$, $C_2^4:A_4\times S_4$, $C_2^6:(C_6\times D_6)$, $C_2\wr C_6:D_6$, $C_2^5.D_6^2$, $C_2^5:(C_6\times S_4)$, $(C_2^3\times D_{12}):S_4$, $(A_4\times C_2^4):S_4$, $C_2^4.(A_4\times S_4)$, $A_4\times C_2^4:S_4$

Order 4032: $S_4\times \GL(3,2)$ x 6, $C_3:D_4\times \PSL(2,7)$ x 4, $C_2\times D_6\times \GL(3,2)$ x 4, $F_8:C_3\times S_4$ x 4, $C_3\times C_2^3:\GL(3,2)$ x 3, $C_4\times S_3\times \GL(3,2)$ x 2, $C_3\times D_4\times \GL(3,2)$ x 2, $D_{12}\times \PSL(2,7)$ x 2, $D_6\times F_8:C_6$ x 2, $(D_6\times F_8):C_6$ x 2, $C_2\times A_4\times \GL(3,2)$ x 2, $D_4\times F_8:C_3^2$, $S_3\times F_8:A_4$, $C_3:S_4\times F_8$, $S_3\times A_4\times F_8$, $S_3\times F_8:C_{12}$, $D_{12}\times F_8:C_3$, $C_3\times S_4\times F_8$, $A_4\times F_8:C_6$

Order 3456: $S_3\times S_4^2$ x 26, $C_2\times A_4^2.D_6$ x 6, $A_4\times D_6\times S_4$ x 6, $D_6^2:S_4$ x 4, $D_6\times \PSOPlus(4,3)$ x 3, $C_6\times S_4^2$ x 3, $C_6:S_4^2$ x 3, $C_2^2:C_6^2:S_4$ x 2, $C_6^2.(C_2^2\times S_4)$ x 2, $C_2^4:(C_6\times S_3^2)$ x 2, $D_6^2:S_4$ x 2, $C_2^4:(C_3^2\times S_4)$ x 2, $C_2^2:C_6^2:S_4$ x 2, $C_2^2:C_6^2:S_4$ x 2, $Q_8:A_4:S_3^2$ x 2, $C_6^2:C_2^2:S_4$ x 2, $D_6^2:S_4$ x 2, $C_3\times Q_8:A_4^2$, $(C_2\times D_6):A_4^2$, $C_3\times C_2^3:A_4^2$, $C_2^2:A_4^2:S_3$, $C_2^4:(C_3^2\times S_4)$, $S_3\times S_4\times \SL(2,3)$, $C_2^2:A_4^2\times S_3$, $C_6^2.(C_2^2\times S_4)$, $C_6^2.(C_2^2\times S_4)$, $C_2^2:A_4^2:S_3$, $C_2^4:C_3^2\times S_4$, $C_6^2:C_2^2:S_4$, $Q_8:A_4\times S_3^2$, $Q_8:A_4:S_3^2$, $C_2^3:A_4\times S_3^2$

Order 3072: $C_3:(C_2^5.C_2^5)$, $D_4\times C_2^4:S_4$, $C_2^4:D_4\times S_4$, $D_4\times C_2^4:S_4$

Order 3024: $C_3\times S_3\times \GL(3,2)$ x 4, $S_3\times F_8:C_3^2$ x 2, $C_3:S_3\times \GL(3,2)$ x 2, $C_{21}:C_6\times S_4$, $(C_3^2\times F_8):C_6$

Order 2688: $C_2^4:\GL(3,2)$ x 5, $C_2\times D_4\times \GL(3,2)$ x 2, $D_4\times F_8:C_6$, $S_3\times D_4\times F_8$, $C_2\times S_4\times F_8$

Order 2304: $C_2^4:D_6^2$ x 24, $C_2^2:S_4^2$ x 12, $C_2^2.S_4^2$ x 12, $S_3\times C_2^4:S_4$ x 12, $C_2^4:D_6^2$ x 8, $(C_2^3\times A_4):S_4$ x 7, $S_4\times C_2^2.D_{12}$ x 6, $(C_2^3\times S_4):D_6$ x 6, $C_2^6:S_3^2$ x 6, $(C_2^2\times C_6\times A_4).C_2^3$ x 6, $(C_2\times \GL(2,\mathbb{Z}/4)).D_6$ x 6, $C_2^3:A_4\times S_4$ x 6, $A_4\times C_2^3:S_4$ x 6, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 6, $(C_2^2\times S_4):D_{12}$ x 6, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $(C_2^3\times A_4):S_4$ x 5, $C_2^3.(D_6\times S_4)$ x 4, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:S_3^2$ x 4, $C_2^4:D_6^2$ x 4, $C_2^3.(D_6\times S_4)$ x 4, $C_2^6:S_3^2$ x 4, $C_2^6:S_3^2$ x 4, $C_2^4:D_6^2$ x 4, $C_2^4:(S_3\times S_4)$ x 4, $C_2^4:(S_3\times S_4)$ x 4, $(C_2^2\times D_{12}):S_4$ x 4, $S_3\times C_2^4:S_4$ x 4, $(C_2^2\times D_{12}):S_4$ x 4, $C_2\wr C_6:S_3$ x 4, $C_2^6:S_3^2$ x 4, $C_2^6:S_3^2$ x 4, $C_2^6:S_3^2$ x 4, $C_3\times C_2\wr D_6$ x 4, $A_4\times C_2^3:S_4$ x 4, $Q_8:A_4\times S_4$ x 4, $C_2^3:C_{12}\times S_4$ x 3, $(C_2^3\times C_{12}):D_{12}$ x 3, $C_3\times C_2^5:D_{12}$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6:S_3^2$ x 3, $C_2^4:C_6\times S_4$ x 3, $S_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6^2$ x 3, $C_2^6:S_3^2$ x 3, $(A_4\times C_2^4):D_6$ x 3, $A_4\times C_2^4:D_6$ x 3, $(C_2^3\times D_{12}).C_{12}$ x 3, $C_2^6:S_3^2$ x 3, $A_4\times C_2^3:D_{12}$ x 3, $A_4\times C_2^4:D_6$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6^2$ x 3, $C_2^3.(D_6\times S_4)$ x 3, $C_2^4.D_6^2$ x 3, $C_2^2\times S_4^2$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_{12}):D_{12}$ x 3, $(C_6\times \GL(2,\mathbb{Z}/4)):C_4$ x 3, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_{12}):D_{12}$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $C_6:\GL(2,\mathbb{Z}/4):C_4$ x 3, $(C_2^4\times C_6):S_4$ x 2, $S_3\times C_2^4:D_{12}$ x 2, $C_2^4:(S_3\times D_{12})$ x 2, $C_2^2:S_4\times D_{12}$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $(C_2^4\times C_{12}):D_6$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.(D_6\times S_4)$ x 2, $C_2^4:(S_3\times D_{12})$ x 2, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:S_3^2$ x 2, $C_2^4:A_4\times D_6$ x 2, $C_2^4:A_4\times D_6$ x 2, $C_4\times C_2^4:S_3^2$ x 2, $C_4\times C_2^4:S_3^2$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $(C_2^4\times C_{12}):D_6$ x 2, $C_2^4:(C_4\times S_3^2)$ x 2, $(C_2^4\times C_6):S_4$ x 2, $C_2^5:(S_3\times A_4)$ x 2, $C_2\wr C_6:S_3$ x 2, $C_2^4.(S_3\times S_4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_6\times C_2^4:S_4$ x 2, $C_2^6:S_3^2$ x 2, $(C_2^4\times C_{12}):D_6$ x 2, $C_3\times C_2^6:D_6$ x 2, $C_6\times C_2^4:S_4$ x 2, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:(S_3\times A_4)$ x 2, $C_2^4:(C_4\times S_3^2)$ x 2, $S_3\times C_2^4:D_{12}$ x 2, $C_2\wr C_6\times S_3$ x 2, $(C_2^3\times A_4):S_4$ x 2, $A_4\times C_2^3:S_4$ x 2, $(C_2^4\times C_6):S_4$, $(C_2^3\times C_{12}):S_4$, $(C_2^3\times C_{12}):S_4$, $C_2^3:S_4\times C_{12}$, $S_3\times C_2^4.S_4$, $C_2^3.(D_6\times S_4)$, $C_2^5:(C_6\times A_4)$, $C_3\times C_2^5:S_4$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^3:S_4\times C_{12}$, $(C_2^3\times C_{12}):S_4$, $C_2^4.(A_4\times D_6)$, $C_3\times C_2^5:S_4$, $S_3\times C_2^5:A_4$, $C_2^4.D_6^2$, $A_4\times D_{12}:C_2^3$, $C_{12}.(C_2^3\times S_4)$, $C_2^4.D_6^2$, $S_3\times C_2^5:A_4$, $(C_2^3\times C_{12}):S_4$, $C_2^3.(D_6\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^4:A_4^2$, $C_2^4.D_6^2$, $C_2^6:(C_6\times S_3)$, $C_2^4:(C_{12}\times D_6)$, $C_2^4.(A_4\times D_6)$, $(C_2^4\times D_{12}):C_6$, $C_2^3:A_4\times D_{12}$, $(C_2^4\times C_6):S_4$, $C_2^6:C_6\times S_3$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_2^5.(C_6\times D_6)$, $C_2^6:C_6^2$, $C_2^4.D_6^2$, $C_3\times \GL(2,\mathbb{Z}/4):C_2^3$, $C_2^4:D_6^2$, $C_2^4:D_6^2$, $C_2^6:C_6^2$, $(C_2^3\times C_{12}):S_4$, $(C_2^3\times C_{12}):S_4$, $D_{12}\times Q_8:A_4$, $C_2^3:A_4\times S_4$, $C_2^4:(C_6\times S_4)$, $C_2^4:A_4^2$, $S_3\times C_2^4.S_4$

Order 2016: $D_6\times \GL(3,2)$ x 12, $F_8:C_6\times S_3$ x 6, $C_2\times C_6\times \GL(3,2)$ x 4, $A_4\times F_8:C_3$ x 4, $A_4\times \GL(3,2)$ x 4, $C_3:C_4\times \GL(3,2)$ x 2, $C_{12}\times \GL(3,2)$ x 2, $F_8:C_6^2$ x 2, $C_3\times F_8:A_4$, $(C_3\times F_8):C_{12}$, $C_3\times A_4\times F_8$, $C_3\times F_8:C_{12}$, $S_3^2\times F_8$

Order 1344: $C_2^3:\GL(3,2)$

Order 1152: $C_2^3:D_6^2$

Order 576: $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$

Order 288: $C_2^3:C_6^2$

Order 192: $D_6\times C_2^4$, $C_2^3\times S_4$

Order 144: $S_3\times S_4$

Order 96: $C_2^4\times C_6$, $C_2^3\times A_4$

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 48: $C_2^2\times D_6$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$

Order 24: $C_2^2\times C_6$, $C_2\times D_6$, $S_4$

Order 12: $C_2\times C_6$, $A_4$

Order 8: $C_2^3$

Order 6: $S_3$

Order 4: $C_2^2$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 193536: $C_2^5.(S_3^2\times \GL(3,2))$ ($C_1$)

Order 96768: $S_3\times A_4\times C_2^3:\GL(3,2)$ ($C_2$), $(C_2^2\times C_6):(S_4\times \GL(3,2))$ ($C_2$), $C_3\times S_4\times C_2^3:\GL(3,2)$ ($C_2$)

Order 48384: $C_2^5:(C_3^2\times \GL(3,2))$ ($C_2^2$)

Order 32256: $S_4\times C_2^3:\GL(3,2)$ ($S_3$), $(D_6\times C_2^4).\GL(3,2)$ ($S_3$)

Order 16128: $C_6\times C_2^4:\GL(3,2)$ ($D_6$), $A_4\times C_2^3:\GL(3,2)$ ($D_6$)

Order 8064: $S_3\times C_2^3:\GL(3,2)$ ($S_4$)

Order 5376: $C_2^5:\GL(3,2)$ ($S_3^2$)

Order 4032: $C_3\times C_2^3:\GL(3,2)$ ($C_2\times S_4$)

Order 1344: $C_2^3:\GL(3,2)$ ($S_3\times S_4$)

Order 1152: $C_2^3:D_6^2$ ($\PSL(2,7)$)

Order 576: $C_2^2\times A_4\times D_6$ ($C_2\times \GL(3,2)$), $C_2^2\times C_6:S_4$ ($C_2\times \GL(3,2)$), $C_2^2\times C_6\times S_4$ ($C_2\times \GL(3,2)$)

Order 288: $C_2^3:C_6^2$ ($C_2^2\times \GL(3,2)$)

Order 192: $D_6\times C_2^4$ ($S_3\times \GL(3,2)$), $C_2^3\times S_4$ ($S_3\times \GL(3,2)$)

Order 144: $S_3\times S_4$ ($C_2^3:\GL(3,2)$)

Order 96: $C_2^4\times C_6$ ($D_6\times \GL(3,2)$), $C_2^3\times A_4$ ($D_6\times \GL(3,2)$)

Order 72: $S_3\times A_4$ ($C_2^4:\GL(3,2)$), $C_3:S_4$ ($C_2^4:\GL(3,2)$), $C_3\times S_4$ ($C_2^4:\GL(3,2)$)

Order 48: $C_2^2\times D_6$ ($S_4\times \GL(3,2)$)

Order 36: $C_3\times A_4$ ($C_2^5:\GL(3,2)$)

Order 32: $C_2^5$ ($S_3^2\times \GL(3,2)$)

Order 24: $C_2^2\times C_6$ ($C_2\times S_4\times \GL(3,2)$), $C_2\times D_6$ ($S_3\times C_2^3:\GL(3,2)$), $S_4$ ($S_3\times C_2^3:\GL(3,2)$)

Order 12: $C_2\times C_6$ ($S_3\times C_2^4:\GL(3,2)$), $A_4$ ($S_3\times C_2^4:\GL(3,2)$)

Order 8: $C_2^3$ ($S_3\times S_4\times \GL(3,2)$)

Order 6: $S_3$ ($S_4\times C_2^3:\GL(3,2)$)

Order 4: $C_2^2$ ($S_3^2\times C_2^3:\GL(3,2)$)

Order 3: $C_3$ ($(C_2^4\times S_4).\GL(3,2)$)

Order 1: $C_1$ ($C_2^5.(S_3^2\times \GL(3,2))$)

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

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

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

Character theory

Complex character table

See the $165 \times 165$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $150 \times 150$ rational character table (warning: may be slow to load).