Abstract group 3072.vj: $(C_2\times C_4^3).S_4$

• Label: 3072.vj
• Order: \( 2^{10} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$(C_2\times C_4^3).S_4$
Order:	\(3072\)\(\medspace = 2^{10} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_7^3:(C_3\times C_6)$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \)
Composition factors:	$C_2$ x 10, $C_3$
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	111	128	784	384	1152	512	3072
Conjugacy classes	1	8	1	39	3	24	4	80
Divisions	1	8	1	27	3	12	2	54
Autjugacy classes	1	6	1	17	2	2	2	31

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	8	4	24	8	20	4	12	0	0	80
Irr. rational chars.	4	6	12	0	10	4	12	2	4	54

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{3}=e^{4}=f^{4}=g^{4}=[a,f]=[b,f]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,9,18,3,11,20,2,10,17,4,12,19)(5,13,23,8,16,22,6,14,24,7,15,21), (1,8,3,6,2,7,4,5) \!\cdots\! \rangle$
Transitive group:	24T6228				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^3.A_4)$ $\,\rtimes\,$ $C_4$	$(C_4^3.A_4)$ $\,\rtimes\,$ $C_4$	$(C_4.Q_8^2)$ $\,\rtimes\,$ $(C_3:C_4)$	$(C_4^3.C_2^2)$ $\,\rtimes\,$ $(C_3:C_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_4^3)$ . $S_4$	$C_4^3$ . $(A_4:C_4)$ (2)	$(C_2^4.C_2^5)$ . $S_3$	$(C_2^3.C_2^4)$ . $S_4$	all 23

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

Homology

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

Subgroups

There are 22063 subgroups in 2087 conjugacy classes, 39 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $(C_2\times C_4^2).S_4$
Commutator:	$G' \simeq$ $(C_2\times C_4^2).A_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4.C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_2\times C_4^3).S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^4.S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^5.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 3072: $(C_2\times C_4^3).S_4$

Order 1536: $C_2^3.C_2^4.C_6.C_2$ x 2, $C_2^3.C_2^4.C_6.C_2$

Order 1024: $C_2^3.C_2^5.C_2^2$

Order 768: $C_2^4.(C_2\times S_4)$ x 2, $C_4\times C_2^3.S_4$, $(C_2^2\times C_4^2).A_4$, $C_4^3.A_4$, $C_4^3.A_4$

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

Order 384: $C_2^4.S_4$ x 4, $C_2^4.(C_2\times A_4)$ x 2, $C_2^4.S_4$ x 2, $(C_2\times C_4^2).A_4$, $C_4^3:C_6$, $C_4^3:C_6$

Order 256: $C_4^2.(C_2\times D_4)$ x 12, $(C_2\times C_4^2).D_4$ x 8, $(C_2\times C_4^2).D_4$ x 4, $C_4^3.C_2^2$ x 3, $C_4^3.C_2^2$ x 3, $C_2^3.C_2^5$ x 2, $C_2^3.C_2^5$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $C_4^3.C_2^2$ x 2, $C_4^2.\OD_{16}$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times D_4).C_4^2$ x 2, $C_4^2.C_4^2$ x 2, $C_4.D_4^2$ x 2, $C_4^2.C_2^4$ x 2, $C_4^2.C_4^2$ x 2, $(C_2\times D_4):\OD_{16}$ x 2, $C_4^2.C_2^4$, $C_4.Q_8^2$, $C_4^3.C_2^2$, $C_2^3.C_2^5$, $C_4^2:\OD_{16}$, $C_4^2:\OD_{16}$, $C_4^3.C_4$, $C_4^3.C_4$, $(C_2\times D_4).C_4^2$, $C_4^3:C_4$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_2\times \OD_{16}:D_4$, $C_4^3:C_2^2$, $C_2^4.C_2^4$

Order 192: $C_2^3.S_4$ x 4, $C_4\wr C_3$ x 3, $C_2^3.(C_2\times A_4)$ x 2, $C_2^3:\SL(2,3)$ x 2, $Q_8.(C_2\times A_4)$ x 2, $C_2^4.A_4$, $A_4:C_4^2$, $C_4^2:C_{12}$, $C_2^4.A_4$

Order 128: $C_4^2:D_4$ x 12, $C_2^3.C_2^4$ x 10, $C_2^3.Q_{16}$ x 8, $C_2^3.D_8$ x 8, $\OD_{16}:D_4$ x 8, $C_4^2.D_4$ x 8, $C_4^2.D_4$ x 8, $C_2^3.C_2^4$ x 8, $C_4^3:C_2$ x 8, $C_4^2:D_4$ x 8, $C_2^3.C_2^4$ x 7, $C_4^2.D_4$ x 6, $C_2^3.C_2^4$ x 6, $C_2.D_4^2$ x 6, $C_4^2:D_4$ x 5, $C_2^3:C_4^2$ x 4, $C_2^4.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.D_4$ x 4, $C_4^2.C_2^3$ x 4, $C_2^3:\OD_{16}$ x 4, $C_2^3.C_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_2.D_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_2.D_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_2^3.C_2^4$ x 4, $C_2^3.C_2^4$ x 4, $C_4^3:C_2$ x 4, $C_4^3:C_2$ x 4, $C_2^3.C_4^2$ x 3, $C_2.Q_8^2$ x 3, $C_4^2.D_4$ x 3, $C_4^2.D_4$ x 3, $C_4^2:C_8$ x 2, $C_4^2:C_8$ x 2, $(C_2\times Q_8):C_8$ x 2, $(C_2\times D_4):C_8$ x 2, $D_4.C_4^2$ x 2, $D_4:C_4^2$ x 2, $C_2^4.C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $D_4.C_4^2$ x 2, $D_4.C_4^2$ x 2, $C_2.Q_8^2$ x 2, $C_4^2:Q_8$ x 2, $C_4^2:D_4$ x 2, $C_2^3.C_2^4$ x 2, $C_4^2:Q_8$ x 2, $C_2^3.C_2^4$ x 2, $C_4^3:C_2$ x 2, $D_4\times C_4^2$ x 2, $C_2\times C_4^3$, $C_8.C_2^4$, $C_4^2.C_2^3$, $C_2^3.C_2^4$, $C_2.Q_8^2$, $C_2^3.C_2^4$

Order 96: $C_4^2:C_6$ x 4, $C_4^2:C_6$ x 2, $Q_8:A_4$ x 2, $C_2^2.S_4$ x 2, $C_2^3:C_{12}$

Order 64: $C_2^3.C_2^3$ x 32, $C_2^3.C_2^3$ x 24, $C_2^2:C_4^2$ x 24, $C_2^3.D_4$ x 20, $C_2^3:D_4$ x 20, $C_4^2:C_2^2$ x 20, $C_4\times \OD_{16}$ x 18, $C_4^2.C_2^2$ x 18, $C_4^2:C_4$ x 16, $C_4^2.C_2^2$ x 16, $C_4:C_4^2$ x 14, $C_2^3:D_4$ x 14, $\OD_{16}:C_4$ x 12, $C_2^3.D_4$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_2^2$ x 11, $C_2^3.C_2^3$ x 10, $C_2^3:Q_8$ x 10, $C_8.C_2^3$ x 10, $C_2^3.C_2^3$ x 8, $C_2^3.C_2^3$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2.C_2^2$ x 8, $C_4^2:C_4$ x 7, $C_4^3$ x 7, $C_2^3.C_2^3$ x 6, $C_4^2:C_4$ x 6, $C_2^3.D_4$ x 6, $C_2^2.\OD_{16}$ x 6, $C_2^3:C_8$ x 6, $D_4:C_2^3$ x 6, $C_4^2:C_2^2$ x 6, $C_2^2\times C_4^2$ x 6, $C_4^2.C_4$ x 6, $C_4.C_4^2$ x 5, $C_4^2:C_4$ x 4, $C_8.C_2^3$ x 4, $C_2^2\times \OD_{16}$ x 4, $C_2^2.C_4^2$ x 4, $C_4^2.C_2^2$ x 4, $C_4^2.C_2^2$ x 4, $C_4^2.C_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2\times C_4\times C_8$ x 3, $C_2^4:C_4$ x 2, $C_2^2:\OD_{16}$ x 2, $C_4:\OD_{16}$ x 2

Order 48: $A_4:C_4$ x 4, $\SL(2,3):C_2$ x 2, $C_4\times A_4$ x 2, $C_4^2:C_3$ x 2, $C_2^2\times A_4$, $C_{12}:C_4$

Order 32: $C_4\times D_4$ x 52, $C_2\times C_4^2$ x 49, $C_2\times \OD_{16}$ x 40, $C_4:D_4$ x 40, $C_2^3:C_4$ x 39, $C_4^2:C_2$ x 36, $C_4^2:C_2$ x 35, $\OD_{16}:C_2$ x 32, $D_4:C_2^2$ x 27, $C_2.C_4^2$ x 26, $C_2^2.D_4$ x 22, $C_2^2\times D_4$ x 17, $C_8:C_4$ x 16, $C_4\wr C_2$ x 16, $C_4:D_4$ x 13, $C_2^3\times C_4$ x 12, $C_4:Q_8$ x 12, $C_4\times C_8$ x 10, $C_2^3:C_4$ x 8, $C_4\times Q_8$ x 8, $C_2^2:C_8$ x 6, $C_2^2\times Q_8$ x 6, $C_2^2\times C_8$ x 6, $C_4:C_8$ x 6

Order 24: $C_2\times A_4$ x 3, $C_6:C_4$ x 2, $\SL(2,3)$ x 2, $C_2\times C_{12}$

Order 16: $C_2^2\times C_4$ x 72, $C_2\times D_4$ x 65, $C_2^2:C_4$ x 63, $C_4^2$ x 57, $\OD_{16}$ x 40, $D_4:C_2$ x 40, $C_4:C_4$ x 36, $C_2\times C_8$ x 32, $C_2\times Q_8$ x 17, $C_2^4$ x 4

Order 12: $C_3:C_4$ x 4, $C_{12}$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 96, $D_4$ x 43, $C_2^3$ x 22, $C_8$ x 12, $Q_8$ x 11

Order 6: $C_6$ x 3

Order 4: $C_4$ x 27, $C_2^2$ x 24

Order 3: $C_3$

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3072: $(C_2\times C_4^3).S_4$

Order 1536: $C_2^3.C_2^4.C_6.C_2$, $C_2^3.C_2^4.C_6.C_2$

Order 1024: $C_2^3.C_2^5.C_2^2$

Order 768: $C_2^4.(C_2\times S_4)$, $C_4\times C_2^3.S_4$, $(C_2^2\times C_4^2).A_4$, $C_4^3.A_4$, $C_4^3.A_4$

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

Order 384: $(C_2\times C_4^2).A_4$, $C_2^4.S_4$, $C_2^4.(C_2\times A_4)$, $C_4^3:C_6$, $C_4^3:C_6$, $C_2^4.S_4$

Order 256: $C_4^2.(C_2\times D_4)$ x 4, $C_2^3.C_2^5$ x 2, $C_4^3.C_2^2$ x 2, $(C_2\times C_4^2).D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_4^3.C_2^2$ x 2, $C_4^3.C_2^2$ x 2, $C_4^2.C_2^4$, $C_4.Q_8^2$, $C_4^3.C_2^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^4.(C_2\times D_4)$, $C_2^4.(C_2\times D_4)$, $C_4^2:\OD_{16}$, $C_4^2:\OD_{16}$, $C_4^2.\OD_{16}$, $(C_2\times C_4^2).D_4$, $C_4^3.C_4$, $C_4^3.C_4$, $(C_2\times D_4).C_4^2$, $C_4^2.C_4^2$, $(C_2\times D_4).C_4^2$, $C_4^3:C_4$, $C_4.D_4^2$, $C_4^3.C_2^2$, $C_4^3.C_2^2$, $C_2\times \OD_{16}:D_4$, $C_4^3:C_2^2$, $C_2^4.C_2^4$, $C_4^2.C_2^4$, $C_4^2.C_4^2$, $(C_2\times D_4):\OD_{16}$

Order 192: $C_4\wr C_3$ x 3, $C_2^4.A_4$, $A_4:C_4^2$, $C_2^3.(C_2\times A_4)$, $C_2^3:\SL(2,3)$, $C_4^2:C_{12}$, $C_2^3.S_4$, $Q_8.(C_2\times A_4)$, $C_2^4.A_4$

Order 128: $C_4^3:C_2$ x 7, $C_2^3.C_2^4$ x 6, $C_4^2:D_4$ x 6, $C_2^3.C_2^4$ x 5, $C_2^3.C_2^4$ x 4, $C_4^3:C_2$ x 4, $C_4^2:D_4$ x 4, $C_4^2.C_2^3$ x 3, $C_2^3.C_2^4$ x 3, $C_2.D_4^2$ x 3, $C_4^2:D_4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_2^3.C_2^4$ x 3, $C_4^3:C_2$ x 3, $C_2^3.Q_{16}$ x 2, $C_2^3.D_8$ x 2, $C_2^3:C_4^2$ x 2, $C_2^4.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2.C_2^3$ x 2, $C_2^3:\OD_{16}$ x 2, $C_4^2.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^3.C_4^2$ x 2, $C_2.Q_8^2$ x 2, $C_2.Q_8^2$ x 2, $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_4^2:Q_8$ x 2, $C_2^3.C_2^4$ x 2, $C_2\times C_4^3$, $\OD_{16}:D_4$, $C_4^2:C_8$, $C_4^2:C_8$, $(C_2\times Q_8):C_8$, $(C_2\times D_4):C_8$, $D_4.C_4^2$, $D_4:C_4^2$, $C_2^4.C_2^3$, $C_2^4.D_4$, $C_4^2.D_4$, $C_4^2.D_4$, $C_8.C_2^4$, $C_4^2.C_2^3$, $C_4^2.D_4$, $C_4^2.D_4$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_2^3.C_2^4$, $D_4.C_4^2$, $D_4.C_4^2$, $C_4^2:Q_8$, $C_4^2:D_4$, $C_2.Q_8^2$, $C_2^3.C_2^4$, $C_4^3:C_2$, $C_2^3.C_2^4$, $D_4\times C_4^2$

Order 96: $C_4^2:C_6$ x 3, $C_4^2:C_6$, $Q_8:A_4$, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 64: $C_2^3.C_2^3$ x 17, $C_2^3.C_2^3$ x 16, $C_2^2:C_4^2$ x 14, $C_4^2:C_2^2$ x 14, $C_2^3.D_4$ x 10, $C_2^3:D_4$ x 10, $C_4^2.C_2^2$ x 10, $C_4:C_4^2$ x 9, $C_4\times \OD_{16}$ x 8, $C_4^2:C_2^2$ x 8, $C_2^3:D_4$ x 8, $C_4^2.C_2^2$ x 8, $C_2^3.C_2^3$ x 7, $C_4^3$ x 7, $C_4^2:C_2^2$ x 7, $C_2^3.D_4$ x 6, $C_2^3.C_2^3$ x 5, $C_2^3:Q_8$ x 5, $C_4^2:C_4$ x 5, $C_4^2:C_2^2$ x 5, $C_2^2\times C_4^2$ x 5, $C_2^3.C_2^3$ x 4, $C_4^2:C_4$ x 4, $C_4^2:C_4$ x 4, $D_4:C_2^3$ x 4, $C_4^2.C_2^2$ x 4, $C_2^3.D_4$ x 3, $C_8.C_2^3$ x 3, $C_4^2.C_2^2$ x 3, $C_4^2:C_4$ x 3, $C_2^3.C_2^3$ x 2, $\OD_{16}:C_4$ x 2, $C_4.C_4^2$ x 2, $C_2\times C_4\times C_8$ x 2, $C_2^2.\OD_{16}$ x 2, $C_2^3:C_8$ x 2, $C_2^2\times \OD_{16}$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2.C_2^2$ x 2, $C_2^4:C_4$, $C_2^2:\OD_{16}$, $C_8.C_2^3$, $C_2^2.C_4^2$, $C_4:\OD_{16}$, $C_4^2:C_2^2$

Order 48: $C_4\times A_4$ x 2, $C_4^2:C_3$ x 2, $C_2^2\times A_4$, $\SL(2,3):C_2$, $A_4:C_4$, $C_{12}:C_4$

Order 32: $C_2\times C_4^2$ x 38, $C_4\times D_4$ x 29, $C_2^3:C_4$ x 21, $D_4:C_2^2$ x 16, $C_4^2:C_2$ x 16, $C_2^2.D_4$ x 15, $C_4^2:C_2$ x 14, $C_2.C_4^2$ x 14, $C_4:D_4$ x 13, $C_2^2\times D_4$ x 11, $C_2\times \OD_{16}$ x 10, $C_2^3\times C_4$ x 9, $C_4:D_4$ x 8, $C_4\times Q_8$ x 8, $C_4:Q_8$ x 6, $C_2^2\times Q_8$ x 4, $C_8:C_4$ x 4, $C_4\times C_8$ x 4, $C_2^2:C_8$ x 2, $\OD_{16}:C_2$ x 2, $C_2^2\times C_8$ x 2, $C_4:C_8$ x 2, $C_4\wr C_2$ x 2, $C_2^3:C_4$

Order 24: $C_2\times A_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $\SL(2,3)$

Order 16: $C_2^2\times C_4$ x 45, $C_4^2$ x 39, $C_2\times D_4$ x 34, $C_2^2:C_4$ x 26, $C_4:C_4$ x 17, $D_4:C_2$ x 13, $C_2\times Q_8$ x 11, $\OD_{16}$ x 6, $C_2\times C_8$ x 5, $C_2^4$ x 3

Order 12: $C_{12}$ x 2, $C_2\times C_6$, $A_4$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 55, $C_2^3$ x 14, $D_4$ x 14, $Q_8$ x 6, $C_8$ x 2

Order 6: $C_6$ x 2

Order 4: $C_4$ x 17, $C_2^2$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3072: $(C_2\times C_4^3).S_4$ ($C_1$)

Order 1536: $C_2^3.C_2^4.C_6.C_2$ ($C_2$) x 2, $C_2^3.C_2^4.C_6.C_2$ ($C_2$)

Order 768: $(C_2^2\times C_4^2).A_4$ ($C_2^2$), $C_4^3.A_4$ ($C_4$), $C_4^3.A_4$ ($C_4$)

Order 512: $C_2^4.C_2^5$ ($S_3$)

Order 384: $(C_2\times C_4^2).A_4$ ($C_2\times C_4$)

Order 256: $C_2^3.C_2^5$ ($D_6$), $C_4.Q_8^2$ ($C_3:C_4$), $C_4^3.C_2^2$ ($C_3:C_4$)

Order 128: $C_2^3.C_2^4$ ($S_4$) x 2, $C_2\times C_4^3$ ($S_4$), $C_2.Q_8^2$ ($C_6:C_4$)

Order 64: $C_2^3.C_2^3$ ($A_4:C_4$) x 2, $C_2^3:Q_8$ ($C_2\times S_4$) x 2, $C_4^3$ ($A_4:C_4$) x 2, $D_4:C_2^3$ ($A_4:C_4$) x 2, $C_2^2\times C_4^2$ ($C_2\times S_4$)

Order 32: $C_2^2\times Q_8$ ($C_2^2.S_4$) x 2, $C_2^3\times C_4$ ($C_2^2:S_4$), $C_2\times C_4^2$ ($C_2^2.S_4$)

Order 16: $C_2^2\times C_4$ ($C_2^3.S_4$) x 2, $C_2^4$ ($C_2^3:S_4$)

Order 8: $C_2^3$ ($Q_8.(C_2\times S_4)$) x 2, $C_2^3$ ($C_2^4.S_4$)

Order 4: $C_2^2$ ($Q_8.A_4:C_4.C_2$), $C_4$ ($(C_2\times C_4^2).S_4$)

Order 2: $C_2$ ($C_4^2:A_4.C_4.C_2$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 3072: $(C_2\times C_4^3).S_4$ ($C_1$)

Order 1536: $C_2^3.C_2^4.C_6.C_2$ ($C_2$), $C_2^3.C_2^4.C_6.C_2$ ($C_2$)

Order 768: $(C_2^2\times C_4^2).A_4$ ($C_2^2$), $C_4^3.A_4$ ($C_4$), $C_4^3.A_4$ ($C_4$)

Order 512: $C_2^4.C_2^5$ ($S_3$)

Order 384: $(C_2\times C_4^2).A_4$ ($C_2\times C_4$)

Order 256: $C_2^3.C_2^5$ ($D_6$), $C_4.Q_8^2$ ($C_3:C_4$), $C_4^3.C_2^2$ ($C_3:C_4$)

Order 128: $C_2\times C_4^3$ ($S_4$), $C_2.Q_8^2$ ($C_6:C_4$), $C_2^3.C_2^4$ ($S_4$)

Order 64: $C_4^3$ ($A_4:C_4$) x 2, $C_2^3.C_2^3$ ($A_4:C_4$), $C_2^3:Q_8$ ($C_2\times S_4$), $D_4:C_2^3$ ($A_4:C_4$), $C_2^2\times C_4^2$ ($C_2\times S_4$)

Order 32: $C_2^2\times Q_8$ ($C_2^2.S_4$), $C_2^3\times C_4$ ($C_2^2:S_4$), $C_2\times C_4^2$ ($C_2^2.S_4$)

Order 16: $C_2^2\times C_4$ ($C_2^3.S_4$) x 2, $C_2^4$ ($C_2^3:S_4$)

Order 8: $C_2^3$ ($C_2^4.S_4$), $C_2^3$ ($Q_8.(C_2\times S_4)$)

Order 4: $C_2^2$ ($Q_8.A_4:C_4.C_2$), $C_4$ ($(C_2\times C_4^2).S_4$)

Order 2: $C_2$ ($C_4^2:A_4.C_4.C_2$)

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

Series

Derived series

$(C_2\times C_4^3).S_4$

$\rhd$

$(C_2\times C_4^2).A_4$

$\rhd$

$C_2.Q_8^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$(C_2\times C_4^3).S_4$

$\rhd$

$C_2^3.C_2^4.C_6.C_2$

$\rhd$

$(C_2^2\times C_4^2).A_4$

$\rhd$

$(C_2\times C_4^2).A_4$

$\rhd$

$C_2.Q_8^2$

$\rhd$

$C_2^2\times Q_8$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2\times C_4^3).S_4$

$\rhd$

$(C_2\times C_4^2).A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

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

Character theory

Complex character table

See the $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.