Abstract group 24576.qf: $C_4^3.C_2^4:S_4$

• Label: 24576.qf
• Order: \( 2^{13} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_4^3.C_2^4:S_4$
Order:	\(24576\)\(\medspace = 2^{13} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3\times C_4^2.C_2^4.C_6.C_2^5$, of order \(393216\)\(\medspace = 2^{17} \cdot 3 \)
Composition factors:	$C_2$ x 13, $C_3$
Derived length:	$4$

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	1055	512	7648	5632	7680	2048	24576
Conjugacy classes	1	26	1	63	7	35	2	135
Divisions	1	26	1	58	5	28	2	121
Autjugacy classes	1	23	1	44	5	18	1	93

Dimension	1	2	3	4	6	8	12	16	24	48
Irr. complex chars.	8	6	24	1	34	8	27	1	24	2	135
Irr. rational chars.	8	4	24	2	20	4	34	3	16	6	121

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
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:	${\langle a, b, c, d, e, f, g, h, i \mid b^{6}=d^{2}=e^{2}=f^{4}=h^{4}=i^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,11,2,12)(3,10,4,9)(5,15)(6,16)(7,14)(8,13)(17,20,18,19)(21,22), (1,19,15,2,20,16) \!\cdots\! \rangle$
Transitive group:	24T13294	24T13401			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
Possibly split product:	$(C_2^7.C_2^3)$ . $S_4$ (3)	$C_2^7$ . $(C_2^3:S_4)$	$C_2^6$ . $(C_2^4:S_4)$	$C_4^3$ . $(C_2^4:S_4)$	all 51

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 66 normal subgroups (64 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_4^2.C_2^4.D_4:S_3$
Commutator:	$G' \simeq$ $C_2^5.C_2^4.C_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4^2$	$G/\Phi \simeq$ $C_2^2\wr S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4.C_2^6.C_2^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4^3.C_2^4:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^8.D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^5.C_2^3$
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 subgroups of index up to 16 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 24576: $C_4^3.C_2^4:S_4$

Order 12288: $(C_2^3\times C_4^3).S_4$, $C_2^4.C_2^5:S_4$, $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^3).S_4$, $C_2^3:D_4^2.S_4$, $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^5.C_2\wr C_6$

Order 8192: $C_2^5.C_2^5.C_2^3$

Order 6144: $(C_2^4\times C_4^2).S_4$ x 2, $C_2^3.C_2^6.C_6.C_2$, $(C_2^2\times C_4^2).D_6:D_4$, $C_4^3.\GL(2,\mathbb{Z}/4)$, $C_4^3.\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_4^2).S_4$, $(C_2^3\times C_4^3):A_4$, $C_4^2.C_2^5:A_4$, $C_2^5.C_2^2\wr C_3$

Order 4096: $C_2.C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^4.C_2^6.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^6.C_2^2$, $C_2^3.C_2^6.C_2^3$

Order 3072: $(C_2\times C_4^3).S_4$ x 2, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2.(C_2^5.S_3)$ x 2, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.C_2^6.S_3$, $C_2^4.(C_4\times A_4):C_4$, $C_2^5.C_2^4.C_6$, $C_2\times C_4^3.S_4$, $C_2\times C_4^3:S_4$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_4^3.(C_2^2\times A_4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_4^3:(C_2^2\times A_4)$, $C_4^3.(C_2\times \SL(2,3))$, $C_2^4:D_4.S_4$, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_4^2:C_2^4:A_4$, $C_2^2:D_4^2.A_4$, $C_2^7.S_4$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^3):C_6$, $(C_2\times C_4^3):S_4$

Order 2048: $C_2.C_2^6.C_2^4$ x 3, $C_2^4.C_2^5.C_2^2$ x 3, $C_2.C_2^6.C_2^4$ x 3, $C_2.C_2^6.C_2^4$ x 3, $C_2^4.C_2^5.C_2^2$ x 3, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_4.C_2^5.C_2^3.C_2$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^3.C_2^4.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^2.C_2^5.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $(C_2^3\times Q_8).C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2$, $(C_2^3\times C_4).C_2^5.C_2$, $C_4.C_2^5.C_2^3.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^3.C_2^6.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_4.C_2^5.C_2^3.C_2$, $C_2^3.C_2^5.C_2^3$, $(C_2^3\times C_4^3).C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $(C_2^3\times C_4^3).C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^6.C_2$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^5.C_2^4$, $C_2^2.C_2^6.C_2^3$, $(C_2^4\times C_4).C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_4.C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^6.C_2^2$, $(C_2^5\times C_4).C_2^4$, $C_4.C_2^5.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_4.C_2^5.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^6.C_2^2$, $C_4.C_2^5.C_2^3.C_2$, $C_2.C_2^6.C_2^3.C_2$, $C_2^4.C_2^5.C_2^2$, $C_4.C_2^5.C_2^3.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^3$, $C_4.C_2^5.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_4.C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2^3$, $(C_2^5\times C_4).C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $(C_2^5\times C_4).C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $(C_2^5\times C_4).C_2^4$, $(C_2^5\times C_4).C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^4.C_2^5.C_2^2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^2$, $(C_2^3\times C_4).C_2^5.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^7.C_2^4$

Order 1536: $C_4^2:\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times C_4^2).S_4$ x 3, $(C_2^2\times C_4^2).S_4$ x 3, $C_2^6.S_4$ x 3, $C_2^6.S_4$ x 3, $(C_2^2\times C_4^2).S_4$ x 3, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^3:S_4$ x 2, $C_2^4.C_2^4.S_3$ x 2, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_4^3.S_4$ x 2, $C_4^3:S_4$ x 2, $C_2^7.A_4$ x 2, $C_4^2:(D_4\times A_4)$ x 2, $C_4^2.(C_2^3\times A_4)$ x 2, $C_4^2.(D_4\times A_4)$ x 2, $C_4^3:(C_2\times A_4)$ x 2, $C_2\times C_4^3:A_4$ x 2, $C_2^7.D_6$, $C_4^2.(C_2^2\times S_4)$, $(C_2^2\times C_4^2):S_4$, $C_2^6.S_4$, $C_2\wr S_4$, $(C_2^2\times C_4^2):S_4$, $C_4^2.(C_2^3\times A_4)$, $C_2^6:(C_2\times A_4)$, $C_2\wr A_4:C_2$, $C_2^3.C_2^2\wr C_3$, $C_2^5.(C_2^2\times A_4)$, $C_2^6.(C_2\times A_4)$, $C_2^7:A_4$, $(C_2^3\times C_4^2):A_4$, $C_4^2.(C_2^2\times \SL(2,3))$, $C_2^7:A_4$, $(C_2\times C_4)\wr C_3$, $C_4^2.(C_2^2\times \SL(2,3))$, $C_4^3.\SL(2,3)$

Order 1024: $C_2^7.C_2^3$, $C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^2$, $C_2^3.C_2^5.C_2^2$

Order 512: $C_2\times C_4^2.C_2^3.C_2$ x 2, $C_2\times C_4^2.C_2^4$ x 2, $C_4^2.C_2^5$ x 2, $C_2^3.C_2^6$, $C_2^3\times C_4^3$, $C_2\times C_2^2.C_2^6$, $D_4^2:C_2^3$

Order 256: $D_4^2:C_2^2$ x 2, $C_2^4\times C_4^2$, $C_4^3:C_2^2$

Order 128: $C_4^2:C_2^3$ x 3, $C_4^2:C_2^3$ x 2, $C_2\times C_4^3$, $C_2^7$, $C_2^3\times C_4^2$

Order 64: $C_4^3$ x 2, $C_2^2\times C_4^2$ x 2, $C_2^6$

Order 32: $C_2\times C_4^2$ x 4, $C_2^5$ x 2

Order 16: $C_4^2$, $C_2^4$

Order 8: $C_2^3$ x 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 subgroups of index up to 16 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 24576: $C_4^3.C_2^4:S_4$

Order 12288: $(C_2^3\times C_4^3).S_4$, $C_2^4.C_2^5:S_4$, $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^3).S_4$, $C_2^3:D_4^2.S_4$, $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^5.C_2\wr C_6$

Order 8192: $C_2^5.C_2^5.C_2^3$

Order 6144: $(C_2^4\times C_4^2).S_4$ x 2, $C_2^3.C_2^6.C_6.C_2$, $(C_2^2\times C_4^2).D_6:D_4$, $C_4^3.\GL(2,\mathbb{Z}/4)$, $C_4^3.\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_4^2).S_4$, $(C_2^3\times C_4^3):A_4$, $C_4^2.C_2^5:A_4$, $C_2^5.C_2^2\wr C_3$

Order 4096: $C_2.C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^4$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^5.C_2^2$, $C_2^5.C_2^4.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^5.C_2^4.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^3.C_2^6.C_2^3$, $C_2^4.C_2^6.C_2^2$, $C_2.C_2^6.C_2^5$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^3$, $C_2^4.C_2^6.C_2^2$, $C_2^3.C_2^6.C_2^3$

Order 3072: $(C_2\times C_4^3).S_4$ x 2, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$ x 2, $C_4^2.(C_2^5.S_3)$ x 2, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.C_2^6.S_3$, $C_2^4.(C_4\times A_4):C_4$, $C_2^5.C_2^4.C_6$, $C_2\times C_4^3.S_4$, $C_2\times C_4^3:S_4$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\times C_4^2.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_4^3.(C_2^2\times A_4)$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $C_4^3:(C_2^2\times A_4)$, $C_4^3.(C_2\times \SL(2,3))$, $C_2^4:D_4.S_4$, $(C_2\times C_4^2).\GL(2,\mathbb{Z}/4)$, $C_4^2:C_2^4:A_4$, $C_2^2:D_4^2.A_4$, $C_2^7.S_4$, $C_2^5.\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_4^3):C_6$, $(C_2\times C_4^3):S_4$

Order 2048: $C_2.C_2^6.C_2^4$ x 3, $C_2^4.C_2^5.C_2^2$ x 3, $C_2.C_2^6.C_2^4$ x 3, $C_2.C_2^6.C_2^4$ x 3, $C_2^4.C_2^5.C_2^2$ x 3, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2.C_2^6.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^4.C_2^3$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^2.C_2^6.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_4.C_2^5.C_2^3.C_2$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^3.C_2^4.C_2^4$ x 2, $C_2^3.C_2^5.C_2^3$ x 2, $C_2^4.C_2^5.C_2^2$ x 2, $C_2^2.C_2^5.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $(C_2^3\times Q_8).C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^6.C_2$, $(C_2^3\times C_4).C_2^5.C_2$, $C_4.C_2^5.C_2^3.C_2$, $C_2^3.C_2^6.C_2^2$, $C_2^3.C_2^6.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^5.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_4.C_2^5.C_2^3.C_2$, $C_2^3.C_2^5.C_2^3$, $(C_2^3\times C_4^3).C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $(C_2^3\times C_4^3).C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^6.C_2$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^5.C_2^4$, $C_2^2.C_2^6.C_2^3$, $(C_2^4\times C_4).C_2^5$, $C_2^4.C_2^4.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_4.C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^6.C_2^2$, $(C_2^5\times C_4).C_2^4$, $C_4.C_2^5.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_4.C_2^5.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^6.C_2^2$, $C_4.C_2^5.C_2^3.C_2$, $C_2.C_2^6.C_2^3.C_2$, $C_2^4.C_2^5.C_2^2$, $C_4.C_2^5.C_2^3.C_2$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^4.C_2^3$, $C_4.C_2^5.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^3.C_2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_4.C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2^3$, $(C_2^5\times C_4).C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^2$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^6.C_2$, $(C_2^5\times C_4).C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^4.C_2^5.C_2^2$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2.C_2^6.C_2^4$, $(C_2^5\times C_4).C_2^4$, $(C_2^5\times C_4).C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^2.C_2^6.C_2^3$, $C_2^2.C_2^6.C_2^3$, $C_2^4.C_2^5.C_2^2$, $C_2^3.C_2^6.C_2^2$, $C_2^5.C_2^5.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^4.C_2^5.C_2^2$, $(C_2^3\times C_4).C_2^5.C_2$, $C_2^4.C_2^4.C_2^3$, $C_2^3.C_2^5.C_2^3$, $C_2.C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^3$, $C_2^7.C_2^4$

Order 1536: $(C_2^2\times C_4^2).S_4$ x 3, $(C_2^2\times C_4^2).S_4$ x 3, $C_2^6.S_4$ x 3, $C_2^6.S_4$ x 3, $(C_2^2\times C_4^2).S_4$ x 3, $C_4^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.C_2^4.S_3$ x 2, $C_2^7.A_4$ x 2, $C_4^2.(C_2^3\times A_4)$ x 2, $C_2\times C_4^3:A_4$ x 2, $C_2^7.D_6$, $C_4^2.(C_2^2\times S_4)$, $(C_2^2\times C_4^2):S_4$, $C_2^6.S_4$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_2\wr S_4$, $(C_2^2\times C_4^2):S_4$, $C_4^3:S_4$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^2.\GL(2,\mathbb{Z}/4)$, $C_4^3.S_4$, $C_4^3:S_4$, $C_4^2.(C_2^3\times A_4)$, $C_2^6:(C_2\times A_4)$, $C_2\wr A_4:C_2$, $C_2^3.C_2^2\wr C_3$, $C_2^5.(C_2^2\times A_4)$, $C_2^6.(C_2\times A_4)$, $C_2^7:A_4$, $(C_2^3\times C_4^2):A_4$, $C_4^2.(C_2^2\times \SL(2,3))$, $C_2^7:A_4$, $C_4^2:(D_4\times A_4)$, $(C_2\times C_4)\wr C_3$, $C_4^2.(D_4\times A_4)$, $C_4^3:(C_2\times A_4)$, $C_4^2.(C_2^2\times \SL(2,3))$, $C_4^3.\SL(2,3)$

Order 1024: $C_2^7.C_2^3$, $C_2^6.C_2^4$, $C_2^3.C_2^5.C_2^2$, $C_2^3.C_2^5.C_2^2$

Order 512: $C_2\times C_4^2.C_2^3.C_2$ x 2, $C_2\times C_4^2.C_2^4$ x 2, $C_4^2.C_2^5$ x 2, $C_2^3.C_2^6$, $C_2^3\times C_4^3$, $C_2\times C_2^2.C_2^6$, $D_4^2:C_2^3$

Order 256: $D_4^2:C_2^2$ x 2, $C_2^4\times C_4^2$, $C_4^3:C_2^2$

Order 128: $C_4^2:C_2^3$ x 3, $C_4^2:C_2^3$ x 2, $C_2\times C_4^3$, $C_2^7$, $C_2^3\times C_4^2$

Order 64: $C_2^2\times C_4^2$ x 2, $C_4^3$, $C_2^6$

Order 32: $C_2\times C_4^2$ x 4, $C_2^5$ x 2

Order 16: $C_4^2$, $C_2^4$

Order 8: $C_2^3$ x 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: $C_4^3.C_2^4:S_4$ ($C_1$)

Order 12288: $(C_2^3\times C_4^3).S_4$ ($C_2$), $C_2^4.C_2^5:S_4$ ($C_2$), $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2^3\times C_4^3).S_4$ ($C_2$), $C_2^3:D_4^2.S_4$ ($C_2$), $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^5.C_2\wr C_6$ ($C_2$)

Order 6144: $(C_2^4\times C_4^2).S_4$ ($C_2^2$) x 2, $C_2^3.C_2^6.C_6.C_2$ ($C_2^2$), $(C_2^4\times C_4^2).S_4$ ($C_2^2$), $(C_2^3\times C_4^3):A_4$ ($C_2^2$), $C_4^2.C_2^5:A_4$ ($C_2^2$), $C_2^5.C_2^2\wr C_3$ ($C_2^2$)

Order 4096: $C_2^4.C_2^6.C_2^2$ ($S_3$)

Order 3072: $C_2^5.C_2^4.C_6$ ($C_2^3$)

Order 2048: $C_2^3.C_2^6.C_2^2$ ($D_6$), $(C_2^3\times C_4^3).C_2^2$ ($D_6$), $C_2^7.C_2^4$ ($D_6$)

Order 1536: $(C_2^3\times C_4^2):A_4$ ($D_4:C_2$)

Order 1024: $C_2^7.C_2^3$ ($S_4$), $C_2^6.C_2^4$ ($C_2\times D_6$), $C_2^3.C_2^5.C_2^2$ ($S_4$), $C_2^3.C_2^5.C_2^2$ ($S_4$)

Order 512: $C_2\times C_4^2.C_2^3.C_2$ ($C_2\times S_4$) x 2, $C_2\times C_4^2.C_2^4$ ($C_2\times S_4$) x 2, $C_4^2.C_2^5$ ($C_2\times S_4$) x 2, $C_2^3.C_2^6$ ($D_4:S_3$), $C_2^3\times C_4^3$ ($C_2\times S_4$), $C_2\times C_2^2.C_2^6$ ($C_2\times S_4$), $D_4^2:C_2^3$ ($C_2\times S_4$)

Order 256: $D_4^2:C_2^2$ ($C_2^2\times S_4$) x 2, $C_2^4\times C_4^2$ ($C_2^2\times S_4$), $C_4^3:C_2^2$ ($C_2^2:S_4$)

Order 128: $C_4^2:C_2^3$ ($D_4:S_4$) x 2, $C_4^2:C_2^3$ ($C_2^3:S_4$) x 2, $C_2\times C_4^3$ ($C_2^3:S_4$), $C_2^7$ ($C_2^3:S_4$), $C_4^2:C_2^3$ ($C_2^3:S_4$), $C_2^3\times C_4^2$ ($D_4:S_4$)

Order 64: $C_4^3$ ($C_2^4:S_4$) x 2, $C_2^6$ ($C_2^4:S_4$), $C_2^2\times C_4^2$ ($C_2^2\wr S_3$), $C_2^2\times C_4^2$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($C_2^5:S_4$), $C_2^5$ ($C_2^6.D_6$), $C_2\times C_4^2$ ($C_2^4:(C_2\times S_4)$), $C_2\times C_4^2$ ($C_2^3:S_4.C_2^2$), $C_2\times C_4^2$ ($C_2^6.D_6$), $C_2\times C_4^2$ ($C_2^6.D_6$)

Order 16: $C_4^2$ ($C_2^4.C_2^4.S_3$), $C_2^4$ ($C_2^6:S_4$)

Order 8: $C_2^3$ ($C_2^5.S_4.C_2^2$), $C_2^3$ ($C_2^5.A_4.C_2^3$), $C_2^3$ ($C_2^8.D_6$)

Order 4: $C_2^2$ ($C_2^5.A_4.D_4.C_2$)

Order 2: $C_2$ ($C_4^2.C_2^4.D_4:S_3$)

Order 1: $C_1$ ($C_4^3.C_2^4:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 24576: $C_4^3.C_2^4:S_4$ ($C_1$)

Order 12288: $(C_2^3\times C_4^3).S_4$ ($C_2$), $C_2^4.C_2^5:S_4$ ($C_2$), $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2^3\times C_4^3).S_4$ ($C_2$), $C_2^3:D_4^2.S_4$ ($C_2$), $(C_2^3\times C_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^5.C_2\wr C_6$ ($C_2$)

Order 6144: $(C_2^4\times C_4^2).S_4$ ($C_2^2$) x 2, $C_2^3.C_2^6.C_6.C_2$ ($C_2^2$), $(C_2^4\times C_4^2).S_4$ ($C_2^2$), $(C_2^3\times C_4^3):A_4$ ($C_2^2$), $C_4^2.C_2^5:A_4$ ($C_2^2$), $C_2^5.C_2^2\wr C_3$ ($C_2^2$)

Order 4096: $C_2^4.C_2^6.C_2^2$ ($S_3$)

Order 3072: $C_2^5.C_2^4.C_6$ ($C_2^3$)

Order 2048: $C_2^3.C_2^6.C_2^2$ ($D_6$), $(C_2^3\times C_4^3).C_2^2$ ($D_6$), $C_2^7.C_2^4$ ($D_6$)

Order 1536: $(C_2^3\times C_4^2):A_4$ ($D_4:C_2$)

Order 1024: $C_2^7.C_2^3$ ($S_4$), $C_2^6.C_2^4$ ($C_2\times D_6$), $C_2^3.C_2^5.C_2^2$ ($S_4$), $C_2^3.C_2^5.C_2^2$ ($S_4$)

Order 512: $C_2\times C_4^2.C_2^3.C_2$ ($C_2\times S_4$) x 2, $C_2\times C_4^2.C_2^4$ ($C_2\times S_4$) x 2, $C_4^2.C_2^5$ ($C_2\times S_4$) x 2, $C_2^3.C_2^6$ ($D_4:S_3$), $C_2^3\times C_4^3$ ($C_2\times S_4$), $C_2\times C_2^2.C_2^6$ ($C_2\times S_4$), $D_4^2:C_2^3$ ($C_2\times S_4$)

Order 256: $D_4^2:C_2^2$ ($C_2^2\times S_4$) x 2, $C_2^4\times C_4^2$ ($C_2^2\times S_4$), $C_4^3:C_2^2$ ($C_2^2:S_4$)

Order 128: $C_4^2:C_2^3$ ($D_4:S_4$) x 2, $C_4^2:C_2^3$ ($C_2^3:S_4$) x 2, $C_2\times C_4^3$ ($C_2^3:S_4$), $C_2^7$ ($C_2^3:S_4$), $C_4^2:C_2^3$ ($C_2^3:S_4$), $C_2^3\times C_4^2$ ($D_4:S_4$)

Order 64: $C_4^3$ ($C_2^4:S_4$), $C_2^6$ ($C_2^4:S_4$), $C_2^2\times C_4^2$ ($C_2^2\wr S_3$), $C_2^2\times C_4^2$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($C_2^5:S_4$), $C_2^5$ ($C_2^6.D_6$), $C_2\times C_4^2$ ($C_2^4:(C_2\times S_4)$), $C_2\times C_4^2$ ($C_2^3:S_4.C_2^2$), $C_2\times C_4^2$ ($C_2^6.D_6$), $C_2\times C_4^2$ ($C_2^6.D_6$)

Order 16: $C_4^2$ ($C_2^4.C_2^4.S_3$), $C_2^4$ ($C_2^6:S_4$)

Order 8: $C_2^3$ ($C_2^5.S_4.C_2^2$), $C_2^3$ ($C_2^5.A_4.C_2^3$), $C_2^3$ ($C_2^8.D_6$)

Order 4: $C_2^2$ ($C_2^5.A_4.D_4.C_2$)

Order 2: $C_2$ ($C_4^2.C_2^4.D_4:S_3$)

Order 1: $C_1$ ($C_4^3.C_2^4:S_4$)

Series

Derived series

$C_4^3.C_2^4:S_4$

$\rhd$

$C_2^5.C_2^4.C_6$

$\rhd$

$C_2^3.C_2^6$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_4^3.C_2^4:S_4$

$\rhd$

$C_2^5.C_2\wr C_6$

$\rhd$

$(C_2^3\times C_4^3):A_4$

$\rhd$

$C_2^5.C_2^4.C_6$

$\rhd$

$(C_2^3\times C_4^2):A_4$

$\rhd$

$C_2^3.C_2^6$

$\rhd$

$C_2^3\times C_4^2$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4^3.C_2^4:S_4$

$\rhd$

$C_2^5.C_2^4.C_6$

$\rhd$

$(C_2^3\times C_4^2):A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 3 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 $135 \times 135$ 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 $121 \times 121$ rational character table.