Abstract group 2304.bcb: $S_4\times C_4.D_{12}$

• Label: 2304.bcb
• Order: \( 2^{8} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$S_4\times C_4.D_{12}$
Order:	\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_4\times C_2^4:S_5$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	279	26	232	342	512	272	640	2304
Conjugacy classes	1	14	3	13	14	16	20	24	105
Divisions	1	14	3	13	14	8	15	7	75
Autjugacy classes	1	10	3	11	12	8	14	7	66

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	16	28	16	16	20	3	6	0	0	105
Irr. rational chars.	8	16	8	14	12	6	8	1	2	75

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$161280$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{6}=d^{2}=e^{2}=f^{8}=[a,e]=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3,2)(4,7,6)(8,15)(12,14), (1,3,2)(4,5,6,7)(8,9,12,13,15,10,14,11), (4,5,7) \!\cdots\! \rangle$
Direct product:	$S_4$ $\, \times\, $ $(C_4.D_{12})$
Semidirect product:	$(\OD_{16}\times S_4)$ $\,\rtimes\,$ $S_3$	$\OD_{16}$ $\,\rtimes\,$ $(S_3\times S_4)$	$(A_4\times \OD_{16})$ $\,\rtimes\,$ $D_6$	$(A_4:\OD_{16})$ $\,\rtimes\,$ $D_6$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{12}\times S_4)$ . $D_4$ (4)	$(C_4\times S_4)$ . $D_{12}$ (2)	$C_{12}$ . $(D_4\times S_4)$ (2)	$C_4$ . $(S_4\times D_{12})$	all 51

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

Homology

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

Subgroups

There are 27368 subgroups in 1980 conjugacy classes, 101 normal (85 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2.D_6^2.C_2$
Commutator:	$G' \simeq$ $C_2^2:C_6^2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2\times C_4$	$G/\Phi \simeq$ $D_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_{24}:C_2^3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $S_4\times C_4.D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $D_6.D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4.D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 2304: $S_4\times C_4.D_{12}$

Order 1152: $C_6.(C_2^3\times A_4).C_2$, $(C_{12}\times S_4).C_2^2$, $C_2^2.D_6^2.C_2$, $A_4.(C_{12}.D_4)$, $A_4.(C_{12}.D_4)$, $A_4.(C_6\times \OD_{16})$, $A_4.(C_{12}.D_4)$

Order 768: $C_{12}.D_4^2$, $S_4\times \OD_{16}:C_2$

Order 576: $S_4\times D_{12}$ x 4, $C_2^2:D_6^2$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_4\times D_{12}$, $C_2\times C_{12}:S_4$, $C_2\times C_{12}\times S_4$, $C_3\times A_4\times \OD_{16}$, $A_4\times C_3:\OD_{16}$, $C_{12}.(C_2\times S_4)$, $C_3\times A_4:\OD_{16}$, $C_{24}:S_4$, $C_{24}\times S_4$, $C_6.(C_4\times S_4)$, $C_3:C_8\times S_4$, $\OD_{16}:S_3^2$

Order 384: $\OD_{16}:S_4$ x 3, $(C_3\times \OD_{16}):D_4$ x 2, $(C_2\times \OD_{16}):D_6$ x 2, $(C_2^2\times C_4).D_{12}$ x 2, $\OD_{16}\times S_4$ x 2, $C_2\times \OD_{16}:D_6$ x 2, $C_2^5.C_{12}$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4^2.D_{12}$, $C_4^2.D_{12}$, $\OD_{16}:D_{12}$, $C_4^2.D_{12}$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_6:D_4^2$, $C_3\times D_4\times \OD_{16}$, $D_4\times C_3:\OD_{16}$

Order 288: $D_6\times S_4$ x 8, $C_{12}\times S_4$ x 4, $D_6:S_4$ x 4, $A_4\times D_{12}$ x 2, $C_{12}:S_4$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6:S_4$ x 2, $C_2^3.C_6^2$, $D_6\times D_{12}$, $C_2\times A_4:C_{12}$, $C_{24}:D_6$, $A_4\times C_{24}$, $S_3\times C_3:\OD_{16}$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $A_4:C_{24}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_2\times C_6\times S_4$

Order 256: $C_4.D_4^2$

Order 192: $\OD_{16}:D_6$ x 16, $D_4\times D_{12}$ x 8, $D_4\times S_4$ x 4, $(C_3\times \OD_{16}):C_4$ x 4, $C_2^3:D_{12}$ x 4, $A_4:\OD_{16}$ x 3, $A_4\times \OD_{16}$ x 3, $C_8:S_4$ x 2, $C_8\times S_4$ x 2, $C_{24}:D_4$ x 2, $C_2^4.C_{12}$ x 2, $(C_2\times C_6):\OD_{16}$ x 2, $C_4^2.D_6$ x 2, $C_2^3\times S_4$ x 2, $C_{12}:C_2^4$ x 2, $C_2^3\times D_{12}$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:D_{12}$ x 2, $C_{24}:C_2^3$ x 2, $C_2^3:D_{12}$ x 2, $C_{12}.C_2^4$ x 2, $C_2\times C_4:D_{12}$ x 2, $C_{24}:D_4$, $D_4\times C_{24}$, $C_{12}:\OD_{16}$, $C_{12}\times \OD_{16}$, $C_{12}:\OD_{16}$, $C_4^2.D_6$, $C_{12}:\OD_{16}$, $C_{12}:\OD_{16}$, $\OD_{16}:D_6$, $C_2^5:C_6$, $C_2^4.D_6$, $C_2\times D_4\times C_{12}$, $C_4^2:D_6$, $C_4^2:D_6$

Order 144: $S_3\times S_4$ x 8, $A_4\times D_6$ x 4, $C_6:S_4$ x 4, $C_6\times S_4$ x 4, $S_3\times D_{12}$ x 4, $C_{12}.C_{12}$ x 2, $D_6^2$ x 2, $C_{12}\times A_4$ x 2, $C_6:D_{12}$ x 2, $A_4:C_{12}$ x 2, $C_{12}.D_6$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_2^2:C_6^2$, $C_6:D_{12}$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_3^2\times \OD_{16}$

Order 128: $\OD_{16}:D_4$ x 4, $\OD_{16}:D_4$ x 2, $C_2^5.C_4$ x 2, $D_4\times \OD_{16}$ x 2, $\OD_{16}:C_2^3$ x 2, $C_4^2.D_4$, $C_4^2.D_4$, $C_2\times D_4^2$

Order 96: $C_2^2\times D_{12}$ x 25, $C_4.D_{12}$ x 20, $D_4\times D_6$ x 17, $C_2^2:D_{12}$ x 16, $C_2^2\times S_4$ x 13, $C_6\times \OD_{16}$ x 9, $C_6:\OD_{16}$ x 9, $D_4\times C_{12}$ x 8, $C_{12}:D_4$ x 8, $C_4:D_{12}$ x 8, $C_2^2.D_{12}$ x 6, $C_4:D_{12}$ x 4, $C_4\times D_{12}$ x 4, $C_2^2:C_{24}$ x 4, $C_{12}.D_4$ x 4, $C_2^3\times D_6$ x 4, $C_2^3:D_6$ x 4, $\GL(2,\mathbb{Z}/4)$ x 4, $C_4:S_4$ x 4, $C_4\times S_4$ x 4, $C_8\times A_4$ x 3, $A_4:C_8$ x 3, $C_4:C_{24}$ x 2, $C_2^3\times A_4$ x 2, $C_2^3\times C_{12}$ x 2, $C_{12}:C_2^3$ x 2, $D_4\times A_4$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^3:C_{12}$ x 2, $C_{12}.C_2^3$ x 2, $S_3\times \OD_{16}$ x 2, $C_{12}:C_8$ x 2, $C_{12}:C_8$, $\OD_{16}:C_6$, $C_8:C_{12}$, $C_4\times C_{24}$, $C_{12}.D_4$, $C_{12}:C_2^3$, $C_2^2.S_4$, $C_2\times C_4:C_{12}$, $C_2\times C_4\times C_{12}$, $C_2^2.D_{12}$, $C_6.C_4^2$

Order 72: $S_3\times D_6$ x 8, $S_3\times C_{12}$ x 4, $C_3:D_{12}$ x 4, $C_6\times D_6$ x 3, $C_3\times S_4$ x 3, $C_6:D_6$ x 2, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3:D_{12}$ x 2, $C_3\times D_{12}$ x 2, $C_3:C_{24}$ x 2, $C_6\times C_{12}$, $C_6:C_{12}$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $\OD_{16}:C_2^2$ x 16, $D_4^2$ x 8, $C_2^2:\OD_{16}$ x 4, $D_4\times C_2^3$ x 4, $C_2^2\times \OD_{16}$ x 4, $\OD_{16}:C_4$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_8:D_4$ x 4, $C_4\times \OD_{16}$ x 2, $C_4^2:C_2^2$ x 2, $C_8:D_4$ x 2, $C_8\times D_4$ x 2, $C_4:\OD_{16}$ x 2, $C_4^2:C_2^2$

Order 48: $C_2\times D_{12}$ x 67, $S_3\times D_4$ x 36, $C_2^2\times D_6$ x 32, $C_2\times S_4$ x 20, $C_6:D_4$ x 18, $C_2^2\times C_{12}$ x 17, $C_3\times \OD_{16}$ x 13, $C_3:\OD_{16}$ x 13, $D_6:C_4$ x 12, $C_6\times D_4$ x 9, $C_4\times D_6$ x 9, $C_2^2:C_{12}$ x 8, $C_6:C_8$ x 6, $C_2^2\times A_4$ x 6, $C_2\times C_{24}$ x 6, $C_4\times A_4$ x 4, $C_4:C_{12}$ x 4, $C_4\times C_{12}$ x 4, $C_2^3\times C_6$ x 2, $C_{24}:C_2$ x 2, $C_6.C_2^3$ x 2, $S_3\times C_8$ x 2, $A_4:C_4$ x 2, $C_{12}:C_4$ x 2

Order 36: $C_6\times S_3$ x 8, $S_3^2$ x 8, $C_6:S_3$ x 4, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 46, $C_2\times \OD_{16}$ x 18, $\OD_{16}:C_2$ x 17, $C_4:D_4$ x 16, $C_2^2\wr C_2$ x 16, $C_4\times D_4$ x 12, $C_2^2:C_8$ x 8, $C_2^3:C_4$ x 8, $C_2^5$ x 4, $C_2^3\times C_4$ x 4, $C_2^2\times C_8$ x 4, $C_4:D_4$ x 4, $C_4:C_8$ x 4, $C_8:C_4$ x 2, $C_4\times C_8$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

Order 24: $C_2\times D_6$ x 89, $D_{12}$ x 38, $C_2\times C_{12}$ x 30, $C_3:D_4$ x 20, $C_2^2\times C_6$ x 14, $C_4\times S_3$ x 12, $C_3\times D_4$ x 12, $C_{24}$ x 7, $S_4$ x 7, $C_3:C_8$ x 7, $C_2\times A_4$ x 6, $C_6:C_4$ x 5

Order 18: $C_3\times S_3$ x 5, $C_3\times C_6$ x 2, $C_3:S_3$ x 2

Order 16: $C_2\times D_4$ x 121, $C_2^4$ x 32, $C_2^2\times C_4$ x 27, $\OD_{16}$ x 20, $C_2^2:C_4$ x 20, $C_2\times C_8$ x 12, $C_4:C_4$ x 6, $C_4^2$ x 4

Order 12: $D_6$ x 68, $C_2\times C_6$ x 26, $C_{12}$ x 15, $C_3:C_4$ x 4, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 88, $D_4$ x 60, $C_2\times C_4$ x 40, $C_8$ x 8

Order 6: $C_6$ x 14, $S_3$ x 13

Order 4: $C_2^2$ x 68, $C_4$ x 13

Order 3: $C_3$ x 3

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2304: $S_4\times C_4.D_{12}$

Order 1152: $C_6.(C_2^3\times A_4).C_2$, $(C_{12}\times S_4).C_2^2$, $C_2^2.D_6^2.C_2$, $A_4.(C_{12}.D_4)$, $A_4.(C_{12}.D_4)$, $A_4.(C_6\times \OD_{16})$, $A_4.(C_{12}.D_4)$

Order 768: $C_{12}.D_4^2$, $S_4\times \OD_{16}:C_2$

Order 576: $S_4\times D_{12}$ x 4, $C_2^2:D_6^2$, $C_2\times A_4\times D_{12}$, $C_2\times C_{12}:S_4$, $C_2\times C_{12}\times S_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_3\times A_4\times \OD_{16}$, $A_4\times C_3:\OD_{16}$, $C_{12}.(C_2\times S_4)$, $C_3\times A_4:\OD_{16}$, $C_{24}:S_4$, $C_{24}\times S_4$, $C_6.(C_4\times S_4)$, $C_3:C_8\times S_4$, $\OD_{16}:S_3^2$

Order 384: $\OD_{16}:S_4$ x 3, $(C_3\times \OD_{16}):D_4$ x 2, $(C_2\times \OD_{16}):D_6$ x 2, $(C_2^2\times C_4).D_{12}$ x 2, $\OD_{16}\times S_4$ x 2, $C_2\times \OD_{16}:D_6$ x 2, $C_2^5.C_{12}$, $C_4.\GL(2,\mathbb{Z}/4)$, $C_4^2.D_{12}$, $C_4^2.D_{12}$, $\OD_{16}:D_{12}$, $C_4^2.D_{12}$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_6:D_4^2$, $C_3\times D_4\times \OD_{16}$, $D_4\times C_3:\OD_{16}$

Order 288: $C_{12}\times S_4$ x 4, $D_6\times S_4$ x 3, $A_4\times D_{12}$ x 2, $C_{12}:S_4$ x 2, $D_6:S_4$ x 2, $C_2^3.C_6^2$, $D_6\times D_{12}$, $C_2\times A_4:C_{12}$, $C_{24}:D_6$, $A_4\times C_{24}$, $S_3\times C_3:\OD_{16}$, $A_4\times C_3:C_8$, $C_{12}.S_4$, $A_4:C_{24}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_{12}.D_{12}$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 256: $C_4.D_4^2$

Order 192: $\OD_{16}:D_6$ x 16, $D_4\times D_{12}$ x 8, $D_4\times S_4$ x 4, $(C_3\times \OD_{16}):C_4$ x 4, $A_4:\OD_{16}$ x 3, $A_4\times \OD_{16}$ x 3, $C_8:S_4$ x 2, $C_8\times S_4$ x 2, $C_{24}:D_4$ x 2, $C_2^4.C_{12}$ x 2, $(C_2\times C_6):\OD_{16}$ x 2, $C_4^2.D_6$ x 2, $C_2^3\times D_{12}$ x 2, $C_2^3:D_{12}$ x 2, $C_{24}:C_2^3$ x 2, $C_2^3:D_{12}$ x 2, $C_{12}.C_2^4$ x 2, $C_2^3:D_{12}$ x 2, $C_{24}:D_4$, $D_4\times C_{24}$, $C_{12}:\OD_{16}$, $C_{12}\times \OD_{16}$, $C_{12}:\OD_{16}$, $C_4^2.D_6$, $C_{12}:\OD_{16}$, $C_{12}:\OD_{16}$, $\OD_{16}:D_6$, $C_2^3\times S_4$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4.D_6$, $C_2\times D_4\times C_{12}$, $C_2\times C_4:D_{12}$, $C_4^2:D_6$, $C_4^2:D_6$

Order 144: $S_3\times D_{12}$ x 4, $C_6\times S_4$ x 3, $C_{12}.C_{12}$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $S_3\times S_4$ x 2, $C_{12}\times A_4$ x 2, $A_4:C_{12}$ x 2, $C_{12}.D_6$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_2^2:C_6^2$, $D_6^2$, $C_6:D_{12}$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_6:D_{12}$, $C_3^2\times \OD_{16}$

Order 128: $\OD_{16}:D_4$ x 4, $\OD_{16}:D_4$ x 2, $C_2^5.C_4$ x 2, $D_4\times \OD_{16}$ x 2, $\OD_{16}:C_2^3$ x 2, $C_4^2.D_4$, $C_4^2.D_4$, $C_2\times D_4^2$

Order 96: $C_2^2\times D_{12}$ x 20, $C_4.D_{12}$ x 19, $C_6\times \OD_{16}$ x 9, $C_6:\OD_{16}$ x 9, $C_2^2:D_{12}$ x 8, $D_4\times C_{12}$ x 8, $C_{12}:D_4$ x 8, $D_4\times D_6$ x 7, $C_2^2\times S_4$ x 6, $C_4:D_{12}$ x 4, $C_4\times D_{12}$ x 4, $C_2^2:C_{24}$ x 4, $C_{12}.D_4$ x 4, $C_4:S_4$ x 4, $C_4\times S_4$ x 4, $C_4:D_{12}$ x 4, $C_8\times A_4$ x 3, $A_4:C_8$ x 3, $C_2^2.D_{12}$ x 3, $C_4:C_{24}$ x 2, $C_2^3\times D_6$ x 2, $C_2^3\times C_{12}$ x 2, $C_2^3:D_6$ x 2, $D_4\times A_4$ x 2, $C_2^3:C_{12}$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2\times C_{24}$ x 2, $C_2^3:C_{12}$ x 2, $C_{12}.C_2^3$ x 2, $S_3\times \OD_{16}$ x 2, $C_{12}:C_8$ x 2, $C_{12}:C_8$, $\OD_{16}:C_6$, $C_8:C_{12}$, $C_4\times C_{24}$, $C_{12}.D_4$, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_{12}:C_2^3$, $C_2^2.S_4$, $C_2\times C_4:C_{12}$, $C_2\times C_4\times C_{12}$, $C_2^2.D_{12}$, $C_6.C_4^2$

Order 72: $S_3\times C_{12}$ x 4, $S_3\times D_6$ x 3, $C_6\times D_6$ x 2, $C_6\times A_4$ x 2, $C_3\times S_4$ x 2, $C_3:D_{12}$ x 2, $C_3\times D_{12}$ x 2, $C_3:D_{12}$ x 2, $C_3:C_{24}$ x 2, $C_6:D_6$, $S_3\times A_4$, $C_3:S_4$, $C_6\times C_{12}$, $C_6:C_{12}$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $\OD_{16}:C_2^2$ x 16, $D_4^2$ x 8, $C_2^2:\OD_{16}$ x 4, $C_2^2\times \OD_{16}$ x 4, $\OD_{16}:C_4$ x 4, $C_8:D_4$ x 4, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_4\times \OD_{16}$ x 2, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_8:D_4$ x 2, $C_8\times D_4$ x 2, $C_4:\OD_{16}$ x 2, $C_4^2:C_2^2$

Order 48: $C_2\times D_{12}$ x 46, $C_2^2\times C_{12}$ x 17, $C_2^2\times D_6$ x 15, $C_3\times \OD_{16}$ x 13, $C_3:\OD_{16}$ x 13, $S_3\times D_4$ x 10, $C_2\times S_4$ x 9, $C_6:D_4$ x 8, $C_2^2:C_{12}$ x 8, $C_6\times D_4$ x 7, $C_6:C_8$ x 6, $C_2\times C_{24}$ x 6, $D_6:C_4$ x 6, $C_2^2\times A_4$ x 4, $C_4\times D_6$ x 4, $C_4\times A_4$ x 4, $C_4:C_{12}$ x 4, $C_4\times C_{12}$ x 4, $C_2^3\times C_6$ x 2, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $A_4:C_4$ x 2, $C_{12}:C_4$ x 2, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 5, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6:S_3$ x 2, $S_3^2$ x 2, $C_6^2$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 29, $C_2\times \OD_{16}$ x 18, $\OD_{16}:C_2$ x 16, $C_4:D_4$ x 12, $C_4\times D_4$ x 12, $C_2^2:C_8$ x 8, $C_2^2\wr C_2$ x 8, $C_2^3:C_4$ x 5, $C_2^2\times C_8$ x 4, $C_4:D_4$ x 4, $C_4:C_8$ x 4, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_8:C_4$ x 2, $C_4\times C_8$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

Order 24: $C_2\times D_6$ x 36, $C_2\times C_{12}$ x 29, $D_{12}$ x 24, $C_2^2\times C_6$ x 12, $C_3\times D_4$ x 8, $C_3:D_4$ x 7, $C_{24}$ x 7, $C_3:C_8$ x 7, $C_4\times S_3$ x 6, $C_2\times A_4$ x 5, $S_4$ x 4, $C_6:C_4$ x 3

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$ x 2, $C_3:S_3$

Order 16: $C_2\times D_4$ x 63, $C_2^2\times C_4$ x 21, $\OD_{16}$ x 20, $C_2^4$ x 16, $C_2^2:C_4$ x 14, $C_2\times C_8$ x 12, $C_4:C_4$ x 6, $C_4^2$ x 4

Order 12: $D_6$ x 28, $C_2\times C_6$ x 21, $C_{12}$ x 14, $C_3:C_4$ x 3, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 41, $D_4$ x 31, $C_2\times C_4$ x 31, $C_8$ x 8

Order 6: $C_6$ x 12, $S_3$ x 7

Order 4: $C_2^2$ x 36, $C_4$ x 11

Order 3: $C_3$ x 3

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2304: $S_4\times C_4.D_{12}$ ($C_1$)

Order 1152: $C_6.(C_2^3\times A_4).C_2$ ($C_2$), $(C_{12}\times S_4).C_2^2$ ($C_2$), $C_2^2.D_6^2.C_2$ ($C_2$), $A_4.(C_{12}.D_4)$ ($C_2$), $A_4.(C_{12}.D_4)$ ($C_2$), $A_4.(C_6\times \OD_{16})$ ($C_2$), $A_4.(C_{12}.D_4)$ ($C_2$)

Order 576: $C_2^2:D_6^2$ ($C_4$) x 2, $C_6:\GL(2,\mathbb{Z}/4)$ ($C_4$) x 2, $C_2\times A_4\times D_{12}$ ($C_2^2$), $C_2\times C_{12}:S_4$ ($C_2^2$), $C_2\times C_{12}\times S_4$ ($C_2^2$), $C_3\times A_4\times \OD_{16}$ ($C_2^2$), $A_4\times C_3:\OD_{16}$ ($C_2^2$), $C_{12}.(C_2\times S_4)$ ($C_2^2$), $C_3\times A_4:\OD_{16}$ ($C_2^2$)

Order 384: $\OD_{16}\times S_4$ ($S_3$), $C_2\times \OD_{16}:D_6$ ($S_3$)

Order 288: $C_{12}\times S_4$ ($D_4$) x 4, $C_2\times A_4\times D_6$ ($C_2\times C_4$) x 2, $C_2\times C_6:S_4$ ($C_2\times C_4$) x 2, $C_2^3.C_6^2$ ($C_2^3$), $C_2\times A_4:C_{12}$ ($C_2\times C_4$), $C_2\times C_6\times S_4$ ($C_2\times C_4$)

Order 192: $A_4:\OD_{16}$ ($D_6$), $C_2^3\times D_{12}$ ($D_6$), $C_2^4.D_6$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $A_4\times \OD_{16}$ ($D_6$)

Order 144: $C_6\times S_4$ ($C_2^2:C_4$) x 2, $C_{12}\times A_4$ ($C_2\times D_4$) x 2, $A_4:C_{12}$ ($C_2^2:C_4$) x 2, $C_2^2:C_6^2$ ($C_2^2\times C_4$)

Order 96: $C_2^3\times D_6$ ($C_4\times S_3$) x 2, $C_4\times S_4$ ($C_3:D_4$) x 2, $C_4\times S_4$ ($D_{12}$) x 2, $C_4.D_{12}$ ($S_4$), $C_2^2\times S_4$ ($C_4\times S_3$), $C_2^3\times C_{12}$ ($C_2\times D_6$), $C_2^3:C_{12}$ ($C_2\times D_6$), $C_2^2.S_4$ ($C_4\times S_3$)

Order 72: $C_3\times S_4$ ($\OD_{16}:C_2$) x 2, $C_6\times A_4$ ($C_2^3:C_4$)

Order 64: $C_2^2\times \OD_{16}$ ($S_3^2$)

Order 48: $C_2\times S_4$ ($D_6:C_4$) x 2, $C_2^2\times C_{12}$ ($S_3\times D_4$) x 2, $A_4:C_4$ ($D_6:C_4$) x 2, $C_2^3\times C_6$ ($C_4\times D_6$), $C_2^2\times A_4$ ($C_4\times D_6$), $C_2\times D_{12}$ ($C_2\times S_4$), $C_4\times A_4$ ($C_6:D_4$), $C_4\times A_4$ ($C_2\times D_{12}$), $C_3\times \OD_{16}$ ($C_2\times S_4$), $C_3:\OD_{16}$ ($C_2\times S_4$)

Order 36: $C_3\times A_4$ ($\OD_{16}:C_2^2$)

Order 32: $C_2^3\times C_4$ ($S_3\times D_6$)

Order 24: $C_2\times D_6$ ($C_4\times S_4$) x 2, $S_4$ ($C_4.D_{12}$) x 2, $C_2\times C_{12}$ ($C_2^2\times S_4$), $C_2^2\times C_6$ ($D_6.D_4$), $C_2\times A_4$ ($C_2^2.D_{12}$)

Order 16: $\OD_{16}$ ($S_3\times S_4$), $C_2^4$ ($C_4\times S_3^2$), $C_2^2\times C_4$ ($D_6:D_6$), $C_2^2\times C_4$ ($S_3\times D_{12}$)

Order 12: $C_{12}$ ($D_4\times S_4$) x 2, $C_2\times C_6$ ($\OD_{16}:D_6$), $C_2\times C_6$ ($C_2^4.D_6$), $A_4$ ($\OD_{16}:D_6$)

Order 8: $C_2^3$ ($D_6.D_{12}$), $C_2\times C_4$ ($D_6\times S_4$)

Order 6: $C_6$ ($C_2^5.D_6$)

Order 4: $C_2^2$ ($C_4\times S_3\times S_4$), $C_2^2$ ($\OD_{16}:S_3^2$), $C_4$ ($C_2^4:S_3^2$), $C_4$ ($S_4\times D_{12}$)

Order 3: $C_3$ ($S_4\times \OD_{16}:C_2$)

Order 2: $C_2$ ($C_2^2.D_6^2.C_2$)

Order 1: $C_1$ ($S_4\times C_4.D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2304: $S_4\times C_4.D_{12}$ ($C_1$)

Order 1152: $C_6.(C_2^3\times A_4).C_2$ ($C_2$), $(C_{12}\times S_4).C_2^2$ ($C_2$), $C_2^2.D_6^2.C_2$ ($C_2$), $A_4.(C_{12}.D_4)$ ($C_2$), $A_4.(C_{12}.D_4)$ ($C_2$), $A_4.(C_6\times \OD_{16})$ ($C_2$), $A_4.(C_{12}.D_4)$ ($C_2$)

Order 576: $C_2^2:D_6^2$ ($C_4$), $C_2\times A_4\times D_{12}$ ($C_2^2$), $C_2\times C_{12}:S_4$ ($C_2^2$), $C_2\times C_{12}\times S_4$ ($C_2^2$), $C_6:\GL(2,\mathbb{Z}/4)$ ($C_4$), $C_3\times A_4\times \OD_{16}$ ($C_2^2$), $A_4\times C_3:\OD_{16}$ ($C_2^2$), $C_{12}.(C_2\times S_4)$ ($C_2^2$), $C_3\times A_4:\OD_{16}$ ($C_2^2$)

Order 384: $\OD_{16}\times S_4$ ($S_3$), $C_2\times \OD_{16}:D_6$ ($S_3$)

Order 288: $C_{12}\times S_4$ ($D_4$) x 4, $C_2^3.C_6^2$ ($C_2^3$), $C_2\times A_4:C_{12}$ ($C_2\times C_4$), $C_2\times A_4\times D_6$ ($C_2\times C_4$), $C_2\times C_6:S_4$ ($C_2\times C_4$), $C_2\times C_6\times S_4$ ($C_2\times C_4$)

Order 192: $A_4:\OD_{16}$ ($D_6$), $C_2^3\times D_{12}$ ($D_6$), $C_2^4.D_6$ ($D_6$), $C_{24}:C_2^3$ ($D_6$), $C_{12}.C_2^4$ ($D_6$), $A_4\times \OD_{16}$ ($D_6$)

Order 144: $C_6\times S_4$ ($C_2^2:C_4$) x 2, $C_{12}\times A_4$ ($C_2\times D_4$) x 2, $A_4:C_{12}$ ($C_2^2:C_4$) x 2, $C_2^2:C_6^2$ ($C_2^2\times C_4$)

Order 96: $C_4\times S_4$ ($C_3:D_4$) x 2, $C_4\times S_4$ ($D_{12}$) x 2, $C_4.D_{12}$ ($S_4$), $C_2^3\times D_6$ ($C_4\times S_3$), $C_2^2\times S_4$ ($C_4\times S_3$), $C_2^3\times C_{12}$ ($C_2\times D_6$), $C_2^3:C_{12}$ ($C_2\times D_6$), $C_2^2.S_4$ ($C_4\times S_3$)

Order 72: $C_6\times A_4$ ($C_2^3:C_4$), $C_3\times S_4$ ($\OD_{16}:C_2$)

Order 64: $C_2^2\times \OD_{16}$ ($S_3^2$)

Order 48: $C_2\times S_4$ ($D_6:C_4$) x 2, $C_2^2\times C_{12}$ ($S_3\times D_4$) x 2, $A_4:C_4$ ($D_6:C_4$) x 2, $C_2^3\times C_6$ ($C_4\times D_6$), $C_2^2\times A_4$ ($C_4\times D_6$), $C_2\times D_{12}$ ($C_2\times S_4$), $C_4\times A_4$ ($C_6:D_4$), $C_4\times A_4$ ($C_2\times D_{12}$), $C_3\times \OD_{16}$ ($C_2\times S_4$), $C_3:\OD_{16}$ ($C_2\times S_4$)

Order 36: $C_3\times A_4$ ($\OD_{16}:C_2^2$)

Order 32: $C_2^3\times C_4$ ($S_3\times D_6$)

Order 24: $C_2\times C_{12}$ ($C_2^2\times S_4$), $C_2^2\times C_6$ ($D_6.D_4$), $C_2\times D_6$ ($C_4\times S_4$), $C_2\times A_4$ ($C_2^2.D_{12}$), $S_4$ ($C_4.D_{12}$)

Order 16: $\OD_{16}$ ($S_3\times S_4$), $C_2^4$ ($C_4\times S_3^2$), $C_2^2\times C_4$ ($D_6:D_6$), $C_2^2\times C_4$ ($S_3\times D_{12}$)

Order 12: $C_{12}$ ($D_4\times S_4$) x 2, $C_2\times C_6$ ($\OD_{16}:D_6$), $C_2\times C_6$ ($C_2^4.D_6$), $A_4$ ($\OD_{16}:D_6$)

Order 8: $C_2^3$ ($D_6.D_{12}$), $C_2\times C_4$ ($D_6\times S_4$)

Order 6: $C_6$ ($C_2^5.D_6$)

Order 4: $C_2^2$ ($C_4\times S_3\times S_4$), $C_2^2$ ($\OD_{16}:S_3^2$), $C_4$ ($C_2^4:S_3^2$), $C_4$ ($S_4\times D_{12}$)

Order 3: $C_3$ ($S_4\times \OD_{16}:C_2$)

Order 2: $C_2$ ($C_2^2.D_6^2.C_2$)

Order 1: $C_1$ ($S_4\times C_4.D_{12}$)

Series

Derived series

$S_4\times C_4.D_{12}$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$S_4\times C_4.D_{12}$

$\rhd$

$C_2^2.D_6^2.C_2$

$\rhd$

$C_2\times C_{12}\times S_4$

$\rhd$

$C_2^3.C_6^2$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_4\times C_4.D_{12}$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2\times C_4$

$\lhd$

$\OD_{16}$

Supergroups

This group is a maximal subgroup of 3 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 $105 \times 105$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $75 \times 75$ rational character table.