Abstract group 36864.i: $C_2^5.S_4\wr C_2$

• Label: 36864.i
• Order: \( 2^{12} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$C_2^5.S_4\wr C_2$
Order:	\(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^6.S_4\wr C_2$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \)
Composition factors:	$C_2$ x 12, $C_3$ x 2
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	1231	1088	12336	9920	6912	5376	36864
Conjugacy classes	1	20	2	35	9	4	5	76
Divisions	1	20	2	35	9	4	5	76
Autjugacy classes	1	18	2	29	8	2	5	65

Dimension	1	2	4	6	9	12	18	24	36	72
Irr. complex chars.	8	2	8	8	8	12	10	4	14	2	76
Irr. rational chars.	8	2	8	8	8	12	10	4	14	2	76

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{6}=d^{6}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,13,6,12,3,14,5,10)(2,15,8,11,4,16,7,9), (1,4,2,3)(5,7,6,8)(10,11)(14,15), (1,10,3,12,2,9)(4,11)(5,13,6,14,8,16)(7,15)\rangle$
Transitive group:	16T1829	16T1831	24T14179	24T14180	all 26
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
Possibly split product:	$C_2^5$ . $(S_4\wr C_2)$	$(C_2^5.S_4^2)$ . $C_2$	$(C_2^5:A_4^2)$ . $D_4$	$(C_2^4.S_4^2)$ . $C_2^2$	all 19
Aut. group:	$\Aut(D_4:C_2^3)$	$\Aut(C_2^8.\SOPlus(4,2))$

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 3076838 subgroups in 29920 conjugacy classes, 24 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^5.S_4\wr C_2$
Commutator:	$G' \simeq$ $C_2^4:\PSOPlus(4,3)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $S_4^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\wr C_2$	$G/\operatorname{Fit} \simeq$ $\SOPlus(4,2)$
Radical:	$R \simeq$ $C_2^5.S_4\wr C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $S_4^2:C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^7.C_2^4.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 128 (order at least 288) are shown, as well as all normal subgroups of any index.

Order 36864: $C_2^5.S_4\wr C_2$

Order 18432: $C_2^8.S_3\wr C_2$ x 2, $C_2^8.\SOPlus(4,2)$ x 2, $C_2^8.C_3:S_3.C_2^2$, $C_2^5.\POPlus(4,3)$, $C_2^5.S_4^2$

Order 9216: $C_2^8.S_3^2$ x 2, $C_2^4.A_4^2:C_4$ x 2, $C_2^8.(C_6\times S_3)$, $C_2^8.S_3^2$, $(C_2^2:S_4)^2$, $C_2^5.(A_4\times S_4)$, $C_2^4.S_4^2$, $C_2^5.\PSOPlus(4,3)$, $C_2^4.S_4^2$

Order 6144: $C_2^5.(D_4\times S_4)$, $C_2^6.(C_2^2\times S_4)$

Order 4608: $C_2^4:\PSOPlus(4,3)$ x 2, $C_2^8:(C_3\times S_3)$ x 2, $C_2^4:(A_4\times S_4)$, $C_2^3:S_4^2$, $C_2^5:A_4^2$, $C_2^6:(C_3\times S_4)$

Order 4096: $C_2^7.C_2^4.C_2$

Order 3072: $C_2^8:D_6$ x 6, $C_2^5:(C_4\times S_4)$ x 2, $C_2^2\wr S_3:D_4$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^8:D_6$ x 2, $C_2^8.C_6.C_2$, $C_2^8.D_6$, $C_2^7:(C_4\times S_3)$, $C_2^7:D_{12}$, $C_2\wr C_6.C_2^3$, $C_2^5:\GL(2,\mathbb{Z}/4)$, $C_2^4.C_2^4:S_3.C_2$, $C_2^7.C_6.C_2^2$

Order 2304: $C_2^2:S_4^2$ x 3, $C_2^2.S_4^2$ x 3, $A_4^2.C_2^3.C_2$, $C_2^4:A_4^2$, $C_2^3:A_4\times S_4$, $A_4\times C_2^3:S_4$, $(C_2^3\times A_4):S_4$

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

Order 1536: $C_2^6:S_4$ x 4, $C_2^7:D_6$ x 4, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^7:D_6$ x 3, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6:D_{12}$ x 3, $C_2^7.D_6$ x 3, $C_2^8:C_6$ x 3, $C_2^7:D_6$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:D_{12}$ x 2, $C_2^7.D_6$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr D_6:C_2$ x 2, $C_2\wr D_6:C_2$ x 2, $C_2^5:(C_2\times S_4)$ x 2, $D_4\times C_2^3:S_4$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^8:C_6$ x 2, $C_2^5:(C_4\times A_4)$ x 2, $C_2^6:S_4$, $C_2^7:D_6$, $C_2^5:(C_2\times S_4)$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2\wr C_2^2\times S_4$, $C_2\wr D_6:C_2$, $C_2\wr D_6:C_2$, $C_2^8:C_6$, $(D_4\times C_2^5):C_6$, $C_2^7:C_{12}$

Order 1152: $S_4\wr C_2$ x 6, $(C_2^2\times A_4):S_4$ x 4, $A_4\times C_2^2:S_4$ x 4, $C_2^2:A_4\times S_4$ x 2, $(C_2^2\times S_4):A_4$ x 2, $A_4^2:C_2^3$, $C_2\times S_4^2$, $C_2\times A_4^2:C_4$, $C_2^3:A_4^2$, $S_3\times C_2^3:S_4$

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

Order 768: $C_2\wr D_6$ x 22, $C_2^5:S_4$ x 11, $C_2^6:D_6$ x 10, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^5:S_4$ x 7, $C_2^6.D_6$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^6:D_6$ x 4, $C_2^6.D_6$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5:D_{12}$ x 4, $C_2^5:S_4$ x 4, $C_2^6.D_6$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^7:C_6$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:S_4$ x 3, $C_2^5:D_{12}$ x 3, $(C_2^2\times D_4):S_4$ x 3, $(C_2^2\times Q_8):S_4$ x 3, $C_2^3:C_4\times S_4$ x 3, $C_2^6:C_{12}$ x 3, $C_2^5:D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $(C_2^3\times C_4):S_4$ x 2, $(C_2^3\times C_4):S_4$ x 2, $(C_2^3\times C_4):S_4$ x 2, $C_4\times C_2^3:S_4$ x 2, $C_2^7:C_6$ x 2, $C_2^6:C_{12}$ x 2, $C_2\wr C_6:C_2$ x 2, $D_4\times C_2^3:A_4$ x 2, $C_2^6.D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times Q_8):S_4$, $(C_2^2\times D_4):S_4$, $C_2^8.C_3$, $C_2^6:A_4$, $C_2^5:(C_2\times A_4)$, $A_4\times C_2\wr C_2^2$, $C_2\wr C_6:C_2$

Order 576: $S_4^2$ x 7, $\POPlus(4,3)$ x 6, $A_4^2:C_4$ x 4, $C_2^2:A_4^2$ x 3, $C_2^4:S_3^2$ x 2, $C_2^4:S_3^2$ x 2, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $S_3\times C_2^3:A_4$, $(C_2^2\times C_6):S_4$, $C_3\times C_2^3:S_4$

Order 512: $C_2^6.D_4$ x 28, $C_2^4.C_2^5$ x 13, $C_2^7:C_2^2$ x 12, $C_2^6:D_4$ x 12, $C_2^4.C_2^5$ x 8, $C_2^6:D_4$ x 7, $C_2^6:C_2^3$ x 6, $C_2^4\wr C_2$ x 6, $C_2^6.D_4$ x 6, $C_2^6.D_4$ x 6, $C_2^6.D_4$ x 6, $D_4^2:D_4$ x 6, $C_2^5.C_2^4$ x 6, $C_2^7.C_2^2$ x 5, $C_2^4.C_2^5$ x 4, $C_2^3:D_4^2$ x 4, $C_2^5.(C_2\times D_4)$ x 4, $C_2^7.C_2^2$ x 4, $C_2^4.C_2^5$ x 4, $C_2^6.D_4$ x 4, $C_2^6:D_4$ x 4, $C_2^6:(C_2\times C_4)$ x 4, $C_2^4.(C_4\times D_4)$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.D_4$ x 4, $D_4^2:D_4$ x 4, $D_4^2:D_4$ x 4, $C_2^7.C_4$ x 4, $C_2^7:C_4$ x 4, $C_2^6.D_4$ x 4, $C_2^7.C_2^2$ x 4, $C_2^6:D_4$ x 4, $C_2^6.D_4$ x 4, $C_2^6.C_2^3$ x 4, $C_2^6.C_2^3$ x 4, $C_2^4.C_2^5$ x 3, $C_2^4.C_2^5$ x 3, $C_2^6:D_4$ x 3, $C_2^6.C_2^3$ x 3, $C_2^5.C_2^4$ x 3, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5:C_2^4$ x 2, $C_2^3:D_4^2$ x 2, $D_4^2:C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5:\SD_{16}$ x 2, $C_2^5:D_8$ x 2, $C_2^5:\OD_{16}$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6:D_4$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^6.D_4$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^3\wr C_2:C_4$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^4.C_2^5$ x 2, $D_4^2:D_4$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $(C_2\times D_4^2):C_4$ x 2, $C_2^6:(C_2\times C_4)$ x 2, $C_2^7:C_4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^4.C_2^5$, $(C_2^2\times C_4^2):C_2^3$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $(C_2^2\times C_4^2):Q_8$, $C_2^6:Q_8$, $(C_2^2\times C_4^2):D_4$, $C_2^6.(C_2\times C_4)$, $C_2^5.(C_2\times D_4)$, $C_2^6.C_2^3$, $D_4^2.C_2^3$, $C_2^6:D_4$

Order 384: $C_2^2\wr S_3$ x 30, $C_2^4:S_4$ x 18, $C_2\wr S_3$ x 16, $C_2^4:S_4$ x 13, $\GL(2,\mathbb{Z}/4):C_2^2$ x 11, $C_2^4:S_4$ x 10, $C_2^5.D_6$ x 10, $C_2\wr C_6$ x 8, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4:D_{12}$ x 8, $C_2^5:A_4$ x 8, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4:D_{12}$ x 8, $C_2^5.D_6$ x 8, $\GL(2,\mathbb{Z}/4):C_2^2$ x 7, $C_2^4.S_4$ x 6, $C_2^5.D_6$ x 6, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:D_{12}$ x 5, $C_2^6:C_6$ x 5, $C_2^5:C_{12}$ x 4, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5:A_4$ x 4, $C_2^6:C_6$ x 4, $C_2^5:C_{12}$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $C_2^5:A_4$ x 3, $C_4\times C_2^3:A_4$ x 2, $C_2^4.S_4$ x 2, $C_2^5.D_6$ x 2, $C_2^4\times S_4$ x 2, $C_2^5:C_{12}$ x 2, $(C_2^2\times Q_8):A_4$, $(C_2^2\times D_4):A_4$, $A_4\times D_4:C_2^2$, $C_2^5.D_6$, $C_2\wr C_2^2\times S_3$

Order 288: $A_4\times S_4$ x 9, $\PSOPlus(4,3)$ x 7, $A_4\wr C_2$ x 6, $(C_2\times C_6):S_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_3\times C_2^3:A_4$, $(C_2\times D_6):A_4$, $S_3\times C_2^2:A_4$, $C_2\times A_4^2$, $D_6\times S_4$

Order 256: $C_2^8$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 288) are shown, as well as all normal subgroups of any index.

Order 36864: $C_2^5.S_4\wr C_2$

Order 18432: $C_2^8.C_3:S_3.C_2^2$, $C_2^8.S_3\wr C_2$, $C_2^8.\SOPlus(4,2)$, $C_2^5.\POPlus(4,3)$, $C_2^5.S_4^2$

Order 9216: $C_2^8.S_3^2$, $C_2^8.(C_6\times S_3)$, $C_2^8.S_3^2$, $C_2^4.A_4^2:C_4$, $(C_2^2:S_4)^2$, $C_2^5.(A_4\times S_4)$, $C_2^4.S_4^2$, $C_2^5.\PSOPlus(4,3)$, $C_2^4.S_4^2$

Order 6144: $C_2^5.(D_4\times S_4)$, $C_2^6.(C_2^2\times S_4)$

Order 4608: $C_2^4:\PSOPlus(4,3)$ x 2, $C_2^4:(A_4\times S_4)$, $C_2^3:S_4^2$, $C_2^5:A_4^2$, $C_2^6:(C_3\times S_4)$, $C_2^8:(C_3\times S_3)$

Order 4096: $C_2^7.C_2^4.C_2$

Order 3072: $C_2^8:D_6$ x 4, $C_2^5:(C_4\times S_4)$ x 2, $C_2^2\wr S_3:D_4$ x 2, $C_2^5:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^8:D_6$ x 2, $C_2^8.C_6.C_2$, $C_2^8.D_6$, $C_2^7:(C_4\times S_3)$, $C_2^7:D_{12}$, $C_2\wr C_6.C_2^3$, $C_2^5:\GL(2,\mathbb{Z}/4)$, $C_2^4.C_2^4:S_3.C_2$, $C_2^7.C_6.C_2^2$

Order 2304: $C_2^2:S_4^2$ x 3, $C_2^2.S_4^2$ x 3, $A_4^2.C_2^3.C_2$, $C_2^4:A_4^2$, $C_2^3:A_4\times S_4$, $A_4\times C_2^3:S_4$, $(C_2^3\times A_4):S_4$

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

Order 1536: $C_2^7:D_6$ x 4, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:S_4$ x 3, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^7:D_6$ x 3, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6:D_{12}$ x 3, $C_2^7.D_6$ x 3, $C_2^7:D_6$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:D_{12}$ x 2, $C_2^7.D_6$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr D_6:C_2$ x 2, $C_2\wr D_6:C_2$ x 2, $C_2^5:(C_2\times S_4)$ x 2, $D_4\times C_2^3:S_4$ x 2, $C_2^4.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^8:C_6$ x 2, $C_2^8:C_6$ x 2, $C_2^5:(C_4\times A_4)$ x 2, $C_2^6:S_4$, $C_2^7:D_6$, $C_2^5:(C_2\times S_4)$, $C_2^4.\GL(2,\mathbb{Z}/4)$, $C_2\wr C_2^2\times S_4$, $C_2\wr D_6:C_2$, $C_2\wr D_6:C_2$, $C_2^8:C_6$, $(D_4\times C_2^5):C_6$, $C_2^7:C_{12}$

Order 1152: $(C_2^2\times A_4):S_4$ x 4, $A_4\times C_2^2:S_4$ x 4, $S_4\wr C_2$ x 3, $C_2^2:A_4\times S_4$ x 2, $(C_2^2\times S_4):A_4$ x 2, $A_4^2:C_2^3$, $C_2\times S_4^2$, $C_2\times A_4^2:C_4$, $C_2^3:A_4^2$, $S_3\times C_2^3:S_4$

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

Order 768: $C_2\wr D_6$ x 22, $C_2^6:D_6$ x 10, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^5:S_4$ x 7, $C_2^5:S_4$ x 7, $C_2^6.D_6$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^6:D_6$ x 4, $C_2^6.D_6$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5:D_{12}$ x 4, $C_2^5:S_4$ x 4, $C_2^6.D_6$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^7:C_6$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:S_4$ x 3, $C_2^5:D_{12}$ x 3, $(C_2^2\times D_4):S_4$ x 3, $(C_2^2\times Q_8):S_4$ x 3, $C_2^3:C_4\times S_4$ x 3, $C_2^6:C_{12}$ x 3, $C_2^5:D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $(C_2^3\times C_4):S_4$ x 2, $(C_2^3\times C_4):S_4$ x 2, $(C_2^3\times C_4):S_4$ x 2, $C_4\times C_2^3:S_4$ x 2, $C_2^7:C_6$ x 2, $C_2^6:C_{12}$ x 2, $C_2\wr C_6:C_2$ x 2, $D_4\times C_2^3:A_4$ x 2, $C_2^6.D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2^2\times Q_8):S_4$, $(C_2^2\times D_4):S_4$, $C_2^8.C_3$, $C_2^6:A_4$, $C_2^5:(C_2\times A_4)$, $A_4\times C_2\wr C_2^2$, $C_2\wr C_6:C_2$

Order 576: $S_4^2$ x 6, $\POPlus(4,3)$ x 4, $C_2^2:A_4^2$ x 3, $C_2^4:S_3^2$ x 2, $A_4^2:C_4$ x 2, $C_2^4:S_3^2$ x 2, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $S_3\times C_2^3:A_4$, $(C_2^2\times C_6):S_4$, $C_3\times C_2^3:S_4$

Order 512: $C_2^6.D_4$ x 21, $C_2^6:D_4$ x 12, $C_2^7:C_2^2$ x 10, $C_2^4.C_2^5$ x 9, $C_2^4.C_2^5$ x 8, $C_2^6:D_4$ x 7, $C_2^5.C_2^4$ x 6, $C_2^6:C_2^3$ x 5, $C_2^4\wr C_2$ x 5, $C_2^4.C_2^5$ x 4, $C_2^3:D_4^2$ x 4, $C_2^7.C_2^2$ x 4, $C_2^7.C_2^2$ x 4, $C_2^6.C_2^3$ x 4, $C_2^6.C_2^3$ x 4, $C_2^4.C_2^5$ x 3, $C_2^4.C_2^5$ x 3, $C_2^6.D_4$ x 3, $C_2^7.C_2^2$ x 3, $C_2^4.C_2^5$ x 3, $C_2^6.D_4$ x 3, $C_2^6.D_4$ x 3, $D_4^2:D_4$ x 3, $C_2^6.C_2^3$ x 3, $C_2^5.C_2^4$ x 3, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5:C_2^4$ x 2, $C_2^3:D_4^2$ x 2, $D_4^2:C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^5:\SD_{16}$ x 2, $C_2^5:D_8$ x 2, $C_2^5:\OD_{16}$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6:D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^6:D_4$ x 2, $C_2^6:D_4$ x 2, $C_2^6:(C_2\times C_4)$ x 2, $C_2^4.(C_4\times D_4)$ x 2, $C_2^6.D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^4.C_2^5$ x 2, $D_4^2:D_4$ x 2, $D_4^2:D_4$ x 2, $C_2^7.C_4$ x 2, $C_2^7:C_4$ x 2, $(C_2\times D_4^2):C_4$ x 2, $C_2^6:(C_2\times C_4)$ x 2, $C_2^6.D_4$ x 2, $C_2^7:C_4$ x 2, $C_2^6:D_4$ x 2, $C_2^6.D_4$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6.C_2^3$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^4.C_2^5$, $(C_2^2\times C_4^2):C_2^3$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^4.C_2^5$, $C_2^5.(C_2\times D_4)$, $(C_2^2\times C_4^2):Q_8$, $C_2^6:Q_8$, $C_2^6.D_4$, $(C_2^2\times C_4^2):D_4$, $C_2^5.(C_2\times D_4)$, $C_2^4.(C_4\times D_4)$, $C_2^3\wr C_2:C_4$, $C_2^4.(C_4\times D_4)$, $C_2^4.(C_4\times D_4)$, $C_2^5.(C_2\times D_4)$, $C_2^6.(C_2\times C_4)$, $D_4^2:D_4$, $C_2^5.(C_2\times D_4)$, $C_2^5.(C_2\times D_4)$, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^5.C_2^4$, $D_4^2.C_2^3$, $C_2^6:D_4$

Order 384: $C_2^2\wr S_3$ x 30, $C_2\wr S_3$ x 16, $C_2^4:S_4$ x 13, $C_2^4:S_4$ x 12, $C_2^4:S_4$ x 10, $C_2^5.D_6$ x 10, $\GL(2,\mathbb{Z}/4):C_2^2$ x 10, $C_2\wr C_6$ x 8, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 8, $C_2^4:D_{12}$ x 8, $\GL(2,\mathbb{Z}/4):C_2^2$ x 7, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 7, $C_2^4:D_{12}$ x 7, $C_2^5.D_6$ x 7, $C_2^4.S_4$ x 6, $C_2^5.D_6$ x 6, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:D_{12}$ x 5, $C_2^5:A_4$ x 5, $C_2^6:C_6$ x 5, $C_2^5:C_{12}$ x 4, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5:A_4$ x 4, $C_2^6:C_6$ x 4, $C_2^5:C_{12}$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $\GL(2,\mathbb{Z}/4):C_2^2$ x 3, $C_2^5:A_4$ x 3, $C_4\times C_2^3:A_4$ x 2, $C_2^4.S_4$ x 2, $C_2^5.D_6$ x 2, $C_2^4\times S_4$ x 2, $C_2^5:C_{12}$ x 2, $(C_2^2\times Q_8):A_4$, $(C_2^2\times D_4):A_4$, $A_4\times D_4:C_2^2$, $C_2^5.D_6$, $C_2\wr C_2^2\times S_3$

Order 288: $A_4\times S_4$ x 8, $\PSOPlus(4,3)$ x 6, $A_4\wr C_2$ x 3, $(C_2\times C_6):S_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_3\times C_2^3:A_4$, $(C_2\times D_6):A_4$, $S_3\times C_2^2:A_4$, $C_2\times A_4^2$, $D_6\times S_4$

Order 256: $C_2^8$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 36864: $C_2^5.S_4\wr C_2$ ($C_1$)

Order 18432: $C_2^8.S_3\wr C_2$ ($C_2$) x 2, $C_2^8.\SOPlus(4,2)$ ($C_2$) x 2, $C_2^8.C_3:S_3.C_2^2$ ($C_2$), $C_2^5.\POPlus(4,3)$ ($C_2$), $C_2^5.S_4^2$ ($C_2$)

Order 9216: $C_2^8.S_3^2$ ($C_2^2$) x 2, $C_2^4.A_4^2:C_4$ ($C_2^2$) x 2, $(C_2^2:S_4)^2$ ($C_2^2$), $C_2^5.\PSOPlus(4,3)$ ($C_2^2$), $C_2^4.S_4^2$ ($C_2^2$)

Order 4608: $C_2^4:\PSOPlus(4,3)$ ($C_2^3$), $C_2^4:\PSOPlus(4,3)$ ($D_4$), $C_2^5:A_4^2$ ($D_4$)

Order 2304: $C_2^4:A_4^2$ ($C_2\times D_4$)

Order 512: $C_2^4\wr C_2$ ($\SOPlus(4,2)$)

Order 256: $C_2^8$ ($S_3^2:C_2^2$)

Order 32: $C_2^5$ ($S_4\wr C_2$)

Order 16: $C_2^4$ ($S_4^2:C_2^2$)

Order 1: $C_1$ ($C_2^5.S_4\wr C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 36864: $C_2^5.S_4\wr C_2$ ($C_1$)

Order 18432: $C_2^8.C_3:S_3.C_2^2$ ($C_2$), $C_2^8.S_3\wr C_2$ ($C_2$), $C_2^8.\SOPlus(4,2)$ ($C_2$), $C_2^5.\POPlus(4,3)$ ($C_2$), $C_2^5.S_4^2$ ($C_2$)

Order 9216: $C_2^8.S_3^2$ ($C_2^2$), $C_2^4.A_4^2:C_4$ ($C_2^2$), $(C_2^2:S_4)^2$ ($C_2^2$), $C_2^5.\PSOPlus(4,3)$ ($C_2^2$), $C_2^4.S_4^2$ ($C_2^2$)

Order 4608: $C_2^4:\PSOPlus(4,3)$ ($C_2^3$), $C_2^4:\PSOPlus(4,3)$ ($D_4$), $C_2^5:A_4^2$ ($D_4$)

Order 2304: $C_2^4:A_4^2$ ($C_2\times D_4$)

Order 512: $C_2^4\wr C_2$ ($\SOPlus(4,2)$)

Order 256: $C_2^8$ ($S_3^2:C_2^2$)

Order 32: $C_2^5$ ($S_4\wr C_2$)

Order 16: $C_2^4$ ($S_4^2:C_2^2$)

Order 1: $C_1$ ($C_2^5.S_4\wr C_2$)

Series

Derived series

$C_2^5.S_4\wr C_2$

$\rhd$

$C_2^4:\PSOPlus(4,3)$

$\rhd$

$C_2^4:A_4^2$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^5.S_4\wr C_2$

$\rhd$

$C_2^5.S_4^2$

$\rhd$

$C_2^5.\PSOPlus(4,3)$

$\rhd$

$C_2^4:\PSOPlus(4,3)$

$\rhd$

$C_2^4:A_4^2$

$\rhd$

$C_2^8$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_2^5.S_4\wr C_2$

$\rhd$

$C_2^4:\PSOPlus(4,3)$

$\rhd$

$C_2^4:A_4^2$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 21 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

See the $76 \times 76$ rational character table. Alternatively, you may search for characters of this group with desired properties.