Abstract group $C_2^7.(D_4\times S_4)$

• Label: 24576.gs
• Order: \( 2^{13} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{17} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: not computed

Group information

Description:	$C_2^7.(D_4\times S_4)$
Order:	\(24576\)\(\medspace = 2^{13} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3:A_4\times S_4$, of order \(393216\)\(\medspace = 2^{17} \cdot 3 \) (generators)
Outer automorphisms:	$C_2^5$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:	$C_2$ x 13, $C_3$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	1183	512	12128	3584	3072	4096	24576
Conjugacy classes	1	47	1	82	5	6	4	146
Divisions	1	47	1	69	5	3	2	128
Autjugacy classes	1	26	1	28	4	3	2	65

Dimension	1	2	3	4	6	12	24	48
Irr. complex chars.	16	12	16	2	28	50	20	2	146
Irr. rational chars.	8	12	8	4	28	44	20	4	128

Minimal Presentations

Permutation degree:	$20$
Transitive degree:	$24$
Rank:	not computed
Inequivalent generating tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{6}=d^{2}=e^{4}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,3,2,6)(4,10,9,5)(7,8,12,13)(11,15,16,14)(17,19,18,20), (1,2,5,11)(3,7,12,6) \!\cdots\! \rangle$
Transitive group:	24T12863	24T13624			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^9.D_4)$ . $S_3$	$(C_2^7.S_4)$ . $D_4$	$C_2^9$ . $(S_3\times D_4)$	$C_2^7$ . $(D_4\times S_4)$	all 52

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 72 normal subgroups (48 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^6.(D_4\times S_4)$
Commutator:	$G' \simeq$ $C_2^8.C_6$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^5:D_4$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9.D_4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^7.(D_4\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^6:(C_2\times S_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^5.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 subgroup of index up to 32 (order at least 768) are shown, as well as all normal subgroups of any index.

Order 24576: $C_2^7.(D_4\times S_4)$

Order 12288: $C_2\times C_2^8.D_6.C_2$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $C_2^7.(D_4\times A_4)$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $C_2^7.C_4:S_4$, $C_2^7.(C_4\times S_4)$, $C_2^8.(C_4\times D_6)$

Order 8192: $C_2^7.C_2^5.C_2$

Order 6144: $C_2^8.D_6.C_2$ x 6, $C_2\times C_2^8.D_6$ x 2, $C_2^9.D_6$ x 2, $C_2^9.D_6$ x 2, $C_2^6.(C_4\times S_4)$ x 2, $C_2\times C_2^8.C_6.C_2$, $C_2^6.(C_4\times S_4)$, $C_2^4\wr C_2.D_6$, $C_2^9.D_6$, $C_2^8.(C_2\times C_{12})$, $C_2^9.C_{12}$

Order 4096: $C_2^7.C_2^4.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^3.C_2^3$ x 2, $C_2^9.D_4$ x 2, $C_2^5.C_2^6.C_2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^5.C_2$, $C_2^8.C_2^4$, $C_2^7.C_2^5$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$

Order 3072: $C_2^8.D_6$ x 8, $C_2^8:D_6$ x 7, $C_2^8.C_6.C_2$ x 3, $C_2^7.S_4$ x 3, $C_2^5.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_2^8.C_6$, $C_2^8.C_{12}$, $C_2^9.C_6$, $C_2^7.S_4$, $C_2^5:(C_4\times S_4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_2^6:(C_4\times D_6)$, $C_2^6:(C_4\times A_4)$, $C_2^6:A_4:C_4$, $C_2^8:C_{12}$, $C_2^7:S_4$, $C_2^7:S_4$

Order 2048: $C_2^5.D_4^2$ x 20, $C_2^8.C_2^3$ x 4, $C_2^7.(C_2\times D_4)$ x 4, $C_2\wr (C_2\times C_4)$ x 4, $C_2^7.(C_2\times D_4)$ x 4, $C_2^5.C_2^5.C_2$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^9.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^7.C_2^4$ x 2, $C_2^5.D_4^2$ x 2, $C_2^8.D_4$ x 2, $C_2^9.C_4$ x 2, $C_2^5.D_4^2$ x 2, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2.C_2^6.C_2^4$, $C_2^8.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^8.C_2^3$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^6.C_2^4.C_2$, $C_2^8.C_2^3$, $C_2^7.C_2^4$

Order 1536: $C_2\times C_2^6.D_6$ x 12, $C_2^6.D_6.C_2$ x 8, $C_2^8.S_3$ x 6, $C_2^6.D_6.C_2$ x 6, $C_2^6.D_6.C_2$ x 4, $C_2\times C_2^6.D_6$ x 2, $C_2^6.C_6.C_2^2$ x 2, $C_2^6.C_6.C_2^2$ x 2, $C_2^6.D_6.C_2$ x 2, $C_2^8.C_6$ x 2, $C_2^8.C_6$ x 2, $C_2\times C_2^6.C_6.C_2$ x 2, $C_2^4:C_3.C_2^4.C_2$, $C_2\times C_2^8.C_3$, $C_2\times C_2^6.C_6.C_2$, $C_2^6.C_{12}.C_2$, $C_2^6.C_6.C_2^2$

Order 1024: $C_2^7:D_4$ x 48, $C_2^4.C_2^6$ x 40, $C_2^6.C_2^4$ x 32, $C_2^6.C_2^4$ x 32, $C_2^6.C_2^4$ x 32, $C_2^6.C_2^4$ x 22, $C_2^2\wr C_2^2$ x 22, $C_2^6.C_2^4$ x 20, $C_2^5.C_2^5$ x 18, $C_2^7:D_4$ x 16, $C_2^7.C_2^3$ x 12, $C_2^7.C_2^3$ x 12, $C_2^7.D_4$ x 12, $C_2^7.C_2^3$ x 11, $C_2^6.C_2^4$ x 10, $C_2^7.D_4$ x 10, $C_2^7:D_4$ x 9, $(C_2^2\wr C_2)^2$ x 9, $C_2^8.C_2^2$ x 7, $C_2^8.C_2^2$ x 7, $C_2^6.C_2^4$ x 6, $C_2^5.C_2^5$ x 6, $C_2^7.D_4$ x 6, $C_2^7:(C_2\times C_4)$ x 6, $C_2^2\wr C_4$ x 5, $C_2^7.C_2^3$ x 4, $C_2^6.C_2^4$ x 4, $C_2^6.C_2^4$ x 4, $C_2^4.C_2^4.C_2^2$ x 4, $C_2^6.C_2^4$ x 4, $C_2^6.C_2^4$ x 4, $C_2^5.C_2^4.C_2$ x 4, $C_2^5.C_2^5$ x 4, $C_2^7.C_2^3$ x 4, $C_2^5.C_2^5$ x 4, $C_2^5.C_2^4.C_2$ x 4, $C_2^5.C_2^5$ x 4, $C_2^4.C_2^5.C_2$ x 4, $C_2^4.C_2^5.C_2$ x 4, $C_2^3.C_2^6.C_2$ x 4, $C_2^7.C_2^3$ x 4, $C_2^6.C_2^4$ x 4, $C_2^5.C_2^5$ x 4, $C_2^9.C_2$ x 4, $C_2^4:D_4^2$ x 4, $C_2^7.(C_2\times C_4)$ x 4, $C_2^6.C_4^2$ x 4, $C_2^7.D_4$ x 4, $C_2^7.C_2^3$ x 4, $C_2^3.C_2^6.C_2$ x 3, $C_2^5.C_2^4.C_2$ x 3, $C_2^5.C_2^5$ x 3, $C_2^4.C_2^5.C_2$ x 3, $C_2^7.C_2^3$ x 3, $C_2^4.C_2^4.C_2^2$ x 3, $C_2^8.C_2^2$ x 3, $C_2^7.C_2^3$ x 3, $C_2^5.C_2^4.C_2$ x 3, $C_2^5.C_2^5$ x 3, $C_2^5.C_2^4.C_2$ x 3, $C_2^4.C_2^5.C_2$ x 3, $C_2.C_2^6.C_2^3$ x 3, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^3.C_2^6.C_2$ x 2, $C_2^7.C_2^3$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^3.C_2^6.C_2$ x 2, $C_2^5.C_2^5$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^9.C_2$ x 2, $C_2^7.C_2^3$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2^6.C_2.C_2^3$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^3.C_2^5.C_2^2$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^5$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^8:C_4$ x 2, $C_2^7:D_4$ x 2, $C_2.C_2^6.C_2^3$, $C_2^5.C_2^5$, $C_2^5.C_2^4.C_2$, $(C_2^4\times C_4).C_2^4$, $C_2.C_2^6.C_2^3$, $C_2^7.C_2^3$, $C_2^4.C_2^4.C_2^2$, $C_2^3.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2$, $C_2^7.C_2^3$, $C_2^7.C_2^3$, $C_2^5.C_2^5$, $C_2^6.C_2^4$, $C_2^5.C_2^4.C_2$, $C_2^7.C_2^3$, $C_2^5.C_2^5$, $C_2^7.C_2^3$, $C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^2$, $C_2^7.C_2^3$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^4.C_2^6$, $C_2^7:D_4$

Order 768: $C_2^5:S_4$ x 47, $C_2^5:S_4$ x 26, $C_2^6:A_4$ x 16, $C_2^5.S_4$ x 6, $C_2^6:A_4$ x 6, $C_2^6:C_{12}$ x 3, $C_2^5:S_4$ x 2, $C_2^5.S_4$ x 2, $C_2^6.D_6$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^3:S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^6:A_4$, $C_2^6:A_4$, $C_2^6:C_{12}$, $C_2^6:C_{12}$

Order 512: $C_2^8.C_2$ x 2, $C_2^6.C_2^3$, $C_2^7.C_2^2$, $C_2^6.C_2^3$, $C_2^9$

Order 256: $C_2^6:C_2^2$ x 2, $C_2^8$, $C_2^5:D_4$

Order 128: $C_2^7$ x 2, $C_2^4:D_4$ x 2

Order 64: $C_2^6$ x 4

Order 32: $C_2^5$ x 3

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroup of index up to 32 (order at least 768) are shown, as well as all normal subgroups of any index.

Order 24576: $C_2^7.(D_4\times S_4)$

Order 12288: $C_2\times C_2^8.D_6.C_2$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $C_2^7.(D_4\times A_4)$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $C_2^7.C_4:S_4$, $C_2^7.(C_4\times S_4)$, $C_2^8.(C_4\times D_6)$

Order 8192: $C_2^7.C_2^5.C_2$

Order 6144: $C_2^8.D_6.C_2$ x 2, $C_2^9.D_6$ x 2, $C_2\times C_2^8.C_6.C_2$, $C_2\times C_2^8.D_6$, $C_2^6.(C_4\times S_4)$, $C_2^4\wr C_2.D_6$, $C_2^9.D_6$, $C_2^9.D_6$, $C_2^8.(C_2\times C_{12})$, $C_2^9.C_{12}$, $C_2^6.(C_4\times S_4)$

Order 4096: $C_2^7.C_2^4.C_2$ x 2, $C_2^7.C_2^4.C_2$ x 2, $C_2^6.C_2^3.C_2^3$ x 2, $C_2^9.D_4$ x 2, $C_2^5.C_2^6.C_2$, $C_2^6.C_2^5.C_2$, $C_2^6.C_2^5.C_2$, $C_2^8.C_2^4$, $C_2^7.C_2^5$, $C_2^7.C_2^4.C_2$, $C_2^6.C_2^5.C_2$

Order 3072: $C_2^8:D_6$ x 4, $C_2^8.C_6.C_2$ x 2, $C_2^7.S_4$ x 2, $C_2^5.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times C_2^8.C_6$, $C_2^8.D_6$, $C_2^8.C_{12}$, $C_2^9.C_6$, $C_2^7.S_4$, $C_2^5:(C_4\times S_4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_2^6:(C_4\times D_6)$, $C_2^6:(C_4\times A_4)$, $C_2^6:A_4:C_4$, $C_2^8:C_{12}$, $C_2^7:S_4$, $C_2^7:S_4$

Order 2048: $C_2^5.D_4^2$ x 6, $C_2^5.C_2^5.C_2$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^8.C_2^3$ x 3, $C_2^5.C_2^5.C_2$ x 3, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^7.C_2^3.C_2$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^5.C_2^5.C_2$ x 2, $C_2^5.C_2^4.C_2^2$ x 2, $C_2^5.C_2^3.C_2^3$ x 2, $C_2^9.C_2^2$ x 2, $C_2^7.C_2^4$ x 2, $C_2^9.C_4$ x 2, $C_2^5.D_4^2$ x 2, $C_2\wr (C_2\times C_4)$ x 2, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^5.C_2^6$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2.C_2^6.C_2^4$, $C_2^7.C_2^4$, $C_2^8.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^8.C_2^3$, $C_2^7.C_2^4$, $C_2^5.C_2^6$, $C_2^5.C_2^5.C_2$, $C_2^7.C_2^4$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^5$, $C_2^5.C_2^5.C_2$, $C_2^6.C_2^4.C_2$, $C_2^7.C_2^4$, $C_2^8.C_2^3$, $C_2^7.C_2^4$, $C_2^5.D_4^2$, $C_2^8.D_4$, $C_2^7.(C_2\times D_4)$, $C_2^7.(C_2\times D_4)$

Order 1536: $C_2\times C_2^6.D_6$ x 7, $C_2^8.S_3$ x 2, $C_2^6.D_6.C_2$ x 2, $C_2^6.C_6.C_2^2$ x 2, $C_2^6.C_6.C_2^2$ x 2, $C_2^6.D_6.C_2$ x 2, $C_2^6.D_6.C_2$ x 2, $C_2^8.C_6$ x 2, $C_2\times C_2^6.C_6.C_2$ x 2, $C_2\times C_2^6.D_6$, $C_2^4:C_3.C_2^4.C_2$, $C_2^6.D_6.C_2$, $C_2\times C_2^8.C_3$, $C_2^8.C_6$, $C_2\times C_2^6.C_6.C_2$, $C_2^6.C_{12}.C_2$, $C_2^6.C_6.C_2^2$

Order 1024: $C_2^2\wr C_2^2$ x 12, $C_2^4.C_2^6$ x 9, $C_2^6.C_2^4$ x 8, $C_2^5.C_2^5$ x 8, $C_2^7.C_2^3$ x 7, $C_2^8.C_2^2$ x 6, $C_2^7:D_4$ x 6, $C_2^7.D_4$ x 6, $C_2^7.D_4$ x 6, $C_2^6.C_2^4$ x 5, $C_2^8.C_2^2$ x 5, $C_2^7:D_4$ x 5, $(C_2^2\wr C_2)^2$ x 5, $C_2^2\wr C_4$ x 5, $C_2^5.C_2^5$ x 4, $C_2^6.C_2^4$ x 4, $C_2^5.C_2^4.C_2$ x 4, $C_2^6.C_2^4$ x 4, $C_2^4.C_2^5.C_2$ x 4, $C_2^3.C_2^6.C_2$ x 4, $C_2^6.C_2^4$ x 4, $C_2^7:D_4$ x 4, $C_2^6.C_2^4$ x 4, $C_2^7.D_4$ x 4, $C_2^7:(C_2\times C_4)$ x 4, $C_2^3.C_2^6.C_2$ x 3, $C_2^5.C_2^4.C_2$ x 3, $C_2^5.C_2^5$ x 3, $C_2^4.C_2^5.C_2$ x 3, $C_2^6.C_2^4$ x 3, $C_2^7.C_2^3$ x 3, $C_2^4.C_2^4.C_2^2$ x 3, $C_2^8.C_2^2$ x 3, $C_2^5.C_2^5$ x 3, $C_2^7.C_2^3$ x 3, $C_2^5.C_2^4.C_2$ x 3, $C_2^5.C_2^5$ x 3, $C_2^5.C_2^5$ x 3, $C_2^5.C_2^4.C_2$ x 3, $C_2^4.C_2^5.C_2$ x 3, $C_2.C_2^6.C_2^3$ x 3, $C_2^9.C_2$ x 3, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^3.C_2^6.C_2$ x 2, $C_2^4.C_2^4.C_2^2$ x 2, $C_2^5.C_2^5$ x 2, $C_2^7.C_2^3$ x 2, $C_2^6.C_2^4$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^5.C_2^5$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^9.C_2$ x 2, $C_2^7.C_2^3$ x 2, $C_2^7.C_2^3$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2$ x 2, $C_2^6.C_2.C_2^3$ x 2, $C_2^6.C_2^4$ x 2, $C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2^7.C_2^3$ x 2, $C_2^7.C_2^3$ x 2, $C_2^5.C_2^4.C_2$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^6.C_2^4$ x 2, $C_2.C_2^6.C_2^3$ x 2, $C_2^7:D_4$ x 2, $C_2^7.(C_2\times C_4)$ x 2, $C_2^6.C_4^2$ x 2, $C_2^7.D_4$ x 2, $C_2^7.C_2^3$ x 2, $C_2.C_2^6.C_2^3$, $C_2^6.C_2^4$, $C_2^5.C_2^5$, $C_2^6.C_2^4$, $C_2^3.C_2^6.C_2$, $C_2^7.C_2^3$, $C_2^5.C_2^4.C_2$, $(C_2^4\times C_4).C_2^4$, $C_2.C_2^6.C_2^3$, $C_2^7.C_2^3$, $C_2^7.C_2^3$, $C_2^4.C_2^4.C_2^2$, $C_2^3.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^7.C_2^3$, $C_2^7.C_2^3$, $C_2^5.C_2^5$, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^6.C_2^4$, $C_2^5.C_2^4.C_2$, $C_2^7.C_2^3$, $C_2^7.C_2^3$, $C_2^5.C_2^5$, $C_2^7.C_2^3$, $C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^2$, $C_2^7.C_2^3$, $C_2^5.C_2^5$, $C_2^3.C_2^5.C_2^2$, $C_2^7.C_2^3$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^5$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^6.C_2^4$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^5$, $C_2^5.C_2^4.C_2$, $C_2^6.C_2^4$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^4.C_2^6$, $C_2^8:C_4$, $C_2^7:D_4$, $C_2^4:D_4^2$

Order 768: $C_2^5:S_4$ x 11, $C_2^6:A_4$ x 10, $C_2^5:S_4$ x 9, $C_2^5.S_4$ x 4, $C_2^6:A_4$ x 4, $C_2^5.S_4$ x 2, $C_2^6.D_6$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:C_{12}$ x 2, $C_2^5:S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $C_4\times C_2^3:S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^6:A_4$, $C_2^6:A_4$, $C_2^6:C_{12}$, $C_2^6:C_{12}$

Order 512: $C_2^8.C_2$ x 2, $C_2^6.C_2^3$, $C_2^7.C_2^2$, $C_2^6.C_2^3$, $C_2^9$

Order 256: $C_2^6:C_2^2$, $C_2^8$, $C_2^5:D_4$

Order 128: $C_2^7$ x 2, $C_2^4:D_4$ x 2

Order 64: $C_2^6$ x 3

Order 32: $C_2^5$ x 3

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 24576: $C_2^7.(D_4\times S_4)$ ($C_1$)

Order 12288: $C_2\times C_2^8.D_6.C_2$ ($C_2$), $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^7.(D_4\times A_4)$ ($C_2$), $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^7.C_4:S_4$ ($C_2$), $C_2^7.(C_4\times S_4)$ ($C_2$), $C_2^8.(C_4\times D_6)$ ($C_2$)

Order 6144: $C_2^8.D_6.C_2$ ($C_4$) x 4, $C_2^9.D_6$ ($C_2^2$) x 2, $C_2\times C_2^8.C_6.C_2$ ($C_2^2$), $C_2^4\wr C_2.D_6$ ($C_2^2$), $C_2^9.D_6$ ($C_2^2$), $C_2^8.(C_2\times C_{12})$ ($C_2^2$), $C_2^9.C_{12}$ ($C_2^2$)

Order 4096: $C_2^9.D_4$ ($S_3$)

Order 3072: $C_2^8:D_6$ ($C_2\times C_4$) x 4, $C_2^8.C_6.C_2$ ($C_2\times C_4$) x 2, $C_2^7.S_4$ ($D_4$) x 2, $C_2^8:D_6$ ($D_4$) x 2, $C_2^9.C_6$ ($C_2^3$)

Order 2048: $C_2^9.C_2^2$ ($D_6$), $C_2^7.C_2^4$ ($D_6$), $C_2^9.C_4$ ($D_6$)

Order 1536: $C_2^8.S_3$ ($C_2^2:C_4$) x 4, $C_2\times C_2^8.C_3$ ($C_2\times D_4$), $C_2^8.C_6$ ($C_2\times D_4$), $C_2^8.C_6$ ($C_2^2\times C_4$)

Order 1024: $(C_2^2\wr C_2)^2$ ($C_4\times S_3$) x 2, $C_2^7.C_2^3$ ($S_4$), $C_2^9.C_2$ ($C_2\times D_6$)

Order 768: $C_2^6:A_4$ ($C_2^3:C_4$)

Order 512: $C_2^6.C_2^3$ ($C_2\times S_4$), $C_2^7.C_2^2$ ($C_2\times S_4$), $C_2^8.C_2$ ($S_3\times D_4$), $C_2^8.C_2$ ($C_4\times D_6$), $C_2^6.C_2^3$ ($C_2\times S_4$), $C_2^9$ ($S_3\times D_4$)

Order 256: $C_2^6:C_2^2$ ($C_4\times S_4$) x 2, $C_2^8$ ($D_6.D_4$), $C_2^5:D_4$ ($C_2^2\times S_4$)

Order 128: $C_2^7$ ($C_2^3:S_4$), $C_2^7$ ($D_4\times S_4$), $C_2^4:D_4$ ($D_4\times S_4$), $C_2^4:D_4$ ($C_2^4.D_6$)

Order 64: $C_2^6$ ($C_2^5.D_6$) x 2, $C_2^6$ ($C_2^4:S_4$), $C_2^6$ ($C_2^5.D_6$)

Order 32: $C_2^5$ ($C_2\wr D_6$) x 2, $C_2^5$ ($C_2^6.D_6$)

Order 16: $C_2^4$ ($C_2^4:C_3.C_2^4.C_2$)

Order 8: $C_2^3$ ($C_2^6:(C_2\times S_4)$)

Order 4: $C_2^2$ ($C_2^6.C_6.C_2^3.C_2$)

Order 2: $C_2$ ($C_2^6.(D_4\times S_4)$)

Order 1: $C_1$ ($C_2^7.(D_4\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24576: $C_2^7.(D_4\times S_4)$ ($C_1$)

Order 12288: $C_2\times C_2^8.D_6.C_2$ ($C_2$), $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^7.(D_4\times A_4)$ ($C_2$), $C_2^7.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^7.C_4:S_4$ ($C_2$), $C_2^7.(C_4\times S_4)$ ($C_2$), $C_2^8.(C_4\times D_6)$ ($C_2$)

Order 6144: $C_2^9.D_6$ ($C_2^2$) x 2, $C_2^8.D_6.C_2$ ($C_4$), $C_2\times C_2^8.C_6.C_2$ ($C_2^2$), $C_2^4\wr C_2.D_6$ ($C_2^2$), $C_2^9.D_6$ ($C_2^2$), $C_2^8.(C_2\times C_{12})$ ($C_2^2$), $C_2^9.C_{12}$ ($C_2^2$)

Order 4096: $C_2^9.D_4$ ($S_3$)

Order 3072: $C_2^8:D_6$ ($C_2\times C_4$) x 2, $C_2^8.C_6.C_2$ ($C_2\times C_4$), $C_2^7.S_4$ ($D_4$), $C_2^9.C_6$ ($C_2^3$), $C_2^8:D_6$ ($D_4$)

Order 2048: $C_2^9.C_2^2$ ($D_6$), $C_2^7.C_2^4$ ($D_6$), $C_2^9.C_4$ ($D_6$)

Order 1536: $C_2^8.S_3$ ($C_2^2:C_4$), $C_2\times C_2^8.C_3$ ($C_2\times D_4$), $C_2^8.C_6$ ($C_2\times D_4$), $C_2^8.C_6$ ($C_2^2\times C_4$)

Order 1024: $C_2^7.C_2^3$ ($S_4$), $C_2^9.C_2$ ($C_2\times D_6$), $(C_2^2\wr C_2)^2$ ($C_4\times S_3$)

Order 768: $C_2^6:A_4$ ($C_2^3:C_4$)

Order 512: $C_2^6.C_2^3$ ($C_2\times S_4$), $C_2^7.C_2^2$ ($C_2\times S_4$), $C_2^8.C_2$ ($S_3\times D_4$), $C_2^8.C_2$ ($C_4\times D_6$), $C_2^6.C_2^3$ ($C_2\times S_4$), $C_2^9$ ($S_3\times D_4$)

Order 256: $C_2^6:C_2^2$ ($C_4\times S_4$), $C_2^8$ ($D_6.D_4$), $C_2^5:D_4$ ($C_2^2\times S_4$)

Order 128: $C_2^7$ ($C_2^3:S_4$), $C_2^7$ ($D_4\times S_4$), $C_2^4:D_4$ ($D_4\times S_4$), $C_2^4:D_4$ ($C_2^4.D_6$)

Order 64: $C_2^6$ ($C_2^5.D_6$), $C_2^6$ ($C_2^4:S_4$), $C_2^6$ ($C_2^5.D_6$)

Order 32: $C_2^5$ ($C_2\wr D_6$) x 2, $C_2^5$ ($C_2^6.D_6$)

Order 16: $C_2^4$ ($C_2^4:C_3.C_2^4.C_2$)

Order 8: $C_2^3$ ($C_2^6:(C_2\times S_4)$)

Order 4: $C_2^2$ ($C_2^6.C_6.C_2^3.C_2$)

Order 2: $C_2$ ($C_2^6.(D_4\times S_4)$)

Order 1: $C_1$ ($C_2^7.(D_4\times S_4)$)

Series

Derived series

$C_2^7.(D_4\times S_4)$

$\rhd$

$C_2^8.C_6$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^7.(D_4\times S_4)$

$\rhd$

$C_2\times C_2^8.D_6.C_2$

$\rhd$

$C_2^9.D_6$

$\rhd$

$C_2^9.C_6$

$\rhd$

$C_2^8.C_6$

$\rhd$

$C_2^6:A_4$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^7.(D_4\times S_4)$

$\rhd$

$C_2^8.C_6$

$\rhd$

$C_2^6:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $146 \times 146$ 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 $128 \times 128$ rational character table (warning: may be slow to load).