Abstract group 6912.hs: $C_2^3:S_4\times S_3^2$

• Label: 6912.hs
• Order: \( 2^{8} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_2^3:S_4\times S_3^2$
Order:	\(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2.A_4^2.C_2^6.C_2^2$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Composition factors:	$C_2$ x 8, $C_3$ x 3
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	12
Elements	1	703	296	1344	2888	1680	6912
Conjugacy classes	1	27	7	16	46	20	117
Divisions	1	27	7	16	46	20	117
Autjugacy classes	1	9	5	6	15	6	42

Dimension	1	2	3	4	6	8	12	16	24	32
Irr. complex chars.	8	12	24	14	28	13	10	6	1	1	117
Irr. rational chars.	8	12	24	14	28	13	10	6	1	1	117

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$188160$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{2}=d^{2}=e^{2}=f^{6}=g^{12}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(7,10)(8,12)(9,13)(11,14), (7,12)(8,10)(9,14)(11,13), (8,13)(9,12), (1,6) \!\cdots\! \rangle$
Direct product:	$S_3$ ${}^2$ $\, \times\, $ $(C_2^3:S_4)$
Semidirect product:	$(D_4:D_6^2)$ $\,\rtimes\,$ $S_3$	$(C_2\times D_6^2)$ $\,\rtimes\,$ $S_4$	$C_2^3$ $\,\rtimes\,$ $(S_4\times S_3^2)$	$(S_3\times Q_8:A_4)$ $\,\rtimes\,$ $D_6$	all 34
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(D_6^2:S_4)$	$D_6$ . $(C_2^4:S_3^2)$	$C_6$ . $(C_2^5:S_3^2)$	$(S_3\times D_6)$ . $(C_2^2:S_4)$	all 7
Aut. group:	$\Aut((C_2\times C_4^2).D_6)$

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 313698 subgroups in 11685 conjugacy classes, 96 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6^2:S_4$
Commutator:	$G' \simeq$ $C_3^2\times Q_8:A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_6^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2\times D_6$
Radical:	$R \simeq$ $C_2^3:S_4\times S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2\wr S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 6912: $C_2^3:S_4\times S_3^2$

Order 3456: $Q_8:A_4:S_3^2$ x 2, $C_6^2:C_2^2:S_4$ x 2, $C_6^2:C_2^2:S_4$, $Q_8:A_4\times S_3^2$, $Q_8:A_4:S_3^2$

Order 2304: $S_3\times C_2^4:S_4$ x 2, $C_2^4:D_6^2$

Order 1728: $A_4:D_6^2$ x 3, $D_4:C_6^2:C_6$ x 2, $(C_2\times C_6^2):S_4$ x 2, $D_4:C_6^2:C_6$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$

Order 1152: $S_3\times C_2^3:S_4$ x 11, $D_6^2:D_4$ x 6, $D_6^2:D_4$ x 6, $D_6^2:D_4$ x 3, $D_6^2:D_4$ x 3, $D_6^2.D_4$ x 3, $D_6^2:D_4$ x 3, $C_2^3.D_6^2$ x 2, $(C_6\times D_{12}):D_4$ x 2, $Q_8:A_4\times D_6$ x 2, $(C_2^3\times C_6):S_4$ x 2, $C_6\times C_2^3:S_4$ x 2, $C_2^3.D_6^2$, $(C_2^2\times C_6^2):C_2^3$, $D_4:D_6^2$

Order 864: $S_4\times S_3^2$ x 24, $S_3\times C_6:S_4$ x 6, $C_6\times S_3\times S_4$ x 6, $D_6^2:C_6$ x 3, $C_2\times A_4:S_3^2$ x 3, $C_6:S_3\times S_4$ x 3, $C_3^2\times Q_8:A_4$, $S_3^2\times \SL(2,3)$

Order 768: $C_2\wr C_2^2\times D_6$ x 2, $C_2^5:S_4$

Order 576: $D_6^2:C_2^2$ x 12, $D_6^2:C_2^2$ x 12, $(C_2\times D_6):D_{12}$ x 12, $D_6^2:C_4$ x 12, $(C_2^2\times C_6):S_4$ x 8, $S_3\times Q_8:A_4$ x 7, $C_3\times C_2^3:S_4$ x 7, $C_2^2:D_6^2$ x 6, $D_6^2:C_2^2$ x 6, $C_4:D_6^2$ x 6, $C_6^2.C_2^4$ x 6, $C_2^4:S_3^2$ x 6, $D_6^2:C_2^2$ x 6, $D_6^2:C_2^2$ x 6, $D_6^2:C_4$ x 6, $C_3\times C_2^4:D_6$ x 6, $S_3\times C_2^3:C_{12}$ x 6, $C_3\times C_2^3:D_{12}$ x 6, $C_2^4:S_3^2$ x 6, $(C_2\times C_{12}):D_{12}$ x 6, $(C_2\times C_{12}):D_{12}$ x 6, $(C_6\times C_{12}):D_4$ x 6, $(C_6\times D_{12}):C_4$ x 6, $(C_2\times D_6):D_{12}$ x 6, $D_6^2:C_4$ x 6, $D_6^2:C_2^2$ x 3, $C_6^2.C_2^4$ x 3, $C_2^4:S_3^2$ x 3, $(C_6\times C_{12}):D_4$ x 3, $(C_6\times D_4).D_6$ x 3, $(C_2\times C_{12}):D_{12}$ x 3, $(C_6\times D_4).D_6$ x 3, $C_6^2.C_2^4$ x 2, $C_6\times Q_8:A_4$ x 2, $D_6^2:C_2^2$ x 2, $(C_6\times C_{12}):D_4$ x 2, $C_2^2\times D_6^2$, $C_6^2.C_2^4$, $D_6^2:C_2^2$, $C_4.D_6^2$, $C_2^4:C_6^2$, $(C_6\times C_{12}):D_4$

Order 432: $A_4:S_3^2$ x 24, $C_6^2:D_6$ x 24, $A_4\times S_3^2$ x 12, $A_4:S_3^2$ x 12, $C_6^2:D_6$ x 12, $C_6\times S_3\times A_4$ x 6, $C_6^2:D_6$ x 6, $C_6:S_3\times A_4$ x 3, $C_6^2:D_6$ x 3, $A_4:C_6^2$ x 3, $C_3\times S_3\times \SL(2,3)$ x 2, $C_2\times S_3^3$, $C_3:S_3\times \SL(2,3)$

Order 384: $C_2\wr C_2^2\times S_3$ x 33, $C_2^4:S_4$ x 8, $C_2^5:D_6$ x 6, $C_2^5:D_6$ x 6, $C_2^3:C_4\times D_6$ x 6, $C_2^4:D_{12}$ x 6, $D_{12}:C_2^4$ x 2, $C_2\wr C_2^2\times C_6$ x 2, $(C_6\times D_4):D_4$ x 2, $C_2^5:A_4$

Order 288: $D_6\times S_4$ x 81, $D_4\times S_3^2$ x 45, $C_2\times D_6^2$ x 27, $D_6^2:C_2$ x 24, $D_6:D_{12}$ x 18, $D_6.D_{12}$ x 18, $C_6^2:C_2^3$ x 12, $D_6^2:C_2$ x 12, $D_{12}:D_6$ x 12, $D_6\times D_{12}$ x 12, $C_6^2:D_4$ x 12, $C_6^2:D_4$ x 12, $C_6^2.C_2^3$ x 12, $C_6^2:D_4$ x 12, $C_6^2:D_4$ x 12, $C_6^2:D_4$ x 12, $C_2^3.S_3^2$ x 12, $C_6^2.D_4$ x 12, $C_6^2.D_4$ x 12, $C_2^3:S_3^2$ x 9, $D_6^2:C_2$ x 9, $D_6:D_{12}$ x 9, $D_6^2.C_2$ x 9, $C_6^2.C_2^3$ x 6, $C_3\times Q_8:A_4$ x 6, $D_{12}:D_6$ x 6, $D_{12}:D_6$ x 6, $D_{12}:D_6$ x 6, $D_{12}:D_6$ x 6, $D_{12}:D_6$ x 6, $C_2.D_6^2$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_6^2.C_2^3$ x 6, $C_6^2:D_4$ x 6, $C_2^3.S_3^2$ x 6, $C_6^2.D_4$ x 6, $C_6^2.D_4$ x 6, $C_6^2.D_4$ x 6, $C_2\times A_4\times D_6$ x 6, $C_2\times C_6:S_4$ x 6, $C_2\times C_6\times S_4$ x 6, $C_6^2:C_2^3$ x 6, $D_4:S_3^2$ x 3, $D_{12}:D_6$ x 3, $D_{12}:D_6$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.D_4$ x 3, $C_6^2.D_4$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2.C_2^3$ x 2, $C_6^2.C_2^3$ x 2, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $D_6\times \SL(2,3)$ x 2, $C_6^2:C_2^3$ x 2, $D_{12}:D_6$, $Q_8\times S_3^2$, $C_6^2:C_2^3$, $D_4:C_6^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 256: $C_2^5:D_4$

Order 216: $C_6^2:C_6$ x 12, $C_3^2:S_4$ x 12, $S_3^3$ x 8, $C_6^2:C_6$ x 6, $C_3^2:S_4$ x 6, $C_3^2\times S_4$ x 6, $C_6^2:C_6$ x 3, $C_6:S_3^2$ x 3, $C_6\times S_3^2$ x 3, $C_6:S_3^2$, $C_3^2\times \SL(2,3)$

Order 192: $C_2^4:D_6$ x 75, $C_2^4:D_6$ x 51, $C_2^3:C_4\times S_3$ x 51, $C_2^3:D_{12}$ x 51, $D_{12}:C_2^3$ x 21, $C_3\times C_2\wr C_2^2$ x 17, $(C_2\times C_{12}):D_4$ x 17, $C_2^3.D_{12}$ x 12, $C_{12}:C_2^4$ x 12, $C_2^4:D_6$ x 12, $C_2^3:D_{12}$ x 12, $C_2^4.D_6$ x 12, $C_2^3:S_4$ x 9, $C_2^4:C_{12}$ x 6, $C_2^4.D_6$ x 6, $D_{12}:C_2^3$ x 6, $C_2^5:C_6$ x 6, $C_2^4:D_6$ x 6, $C_2^4:A_4$ x 5, $C_2^3\times S_4$ x 3, $D_6\times C_2^4$ x 2, $C_{12}.C_2^4$ x 2, $D_{12}:C_2^3$ x 2, $D_{12}:C_2^3$ x 2

Order 144: $D_6^2$ x 193, $S_3\times S_4$ x 132, $D_6:D_6$ x 84, $C_6:S_4$ x 48, $A_4\times D_6$ x 45, $C_6\times S_4$ x 45, $C_{12}:D_6$ x 42, $D_6:D_6$ x 42, $S_3\times D_{12}$ x 42, $C_6^2:C_2^2$ x 30, $C_6^2:C_2^2$ x 24, $C_6:D_{12}$ x 24, $C_4\times S_3^2$ x 22, $C_{12}:D_6$ x 21, $C_{12}:D_6$ x 21, $D_6:C_{12}$ x 18, $C_6.D_{12}$ x 18, $C_6^2:C_2^2$ x 15, $C_6^2:C_4$ x 12, $C_6^2:C_2^2$ x 12, $C_6\times D_{12}$ x 12, $C_{12}\times D_6$ x 12, $D_6:D_6$ x 12, $D_6.D_6$ x 12, $D_6.D_6$ x 12, $C_6.D_{12}$ x 9, $C_6.D_{12}$ x 9, $S_3\times \SL(2,3)$ x 7, $C_2^2:C_6^2$ x 6, $C_4:C_6^2$ x 6, $C_6:D_{12}$ x 6, $C_{12}:D_6$ x 6, $C_{12}.D_6$ x 6, $D_{12}:C_6$ x 6, $C_2^2.S_3^2$ x 6, $D_6.D_6$ x 6, $C_{12}.D_6$ x 6, $C_{12}.D_6$ x 6, $D_{12}:S_3$ x 6, $D_{12}:S_3$ x 6, $C_6^2:C_4$ x 6, $C_6^2:C_4$ x 6, $C_{12}.D_6$ x 3, $C_{12}.D_6$ x 3, $D_{12}:C_6$ x 2, $C_{12}.D_6$ x 2, $C_6\times \SL(2,3)$ x 2, $C_{12}.D_6$ x 2, $C_2^2\times C_6^2$, $C_4.C_6^2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$

Order 128: $C_2^4:D_4$ x 24, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $D_4:C_2^4$

Order 108: $C_3:S_3^2$ x 12, $C_3\times S_3^2$ x 12, $C_3:S_3^2$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3^2\times A_4$ x 3, $C_3^2:D_6$

Order 96: $D_4\times D_6$ x 192, $C_2^2:D_{12}$ x 90, $D_6.D_4$ x 90, $D_6:D_4$ x 90, $D_4:D_6$ x 63, $C_2^3\times D_6$ x 59, $C_2^2.D_{12}$ x 54, $C_2^2\times S_4$ x 48, $C_2^4:C_6$ x 39, $C_2^4:S_3$ x 39, $D_4:D_6$ x 33, $D_4:D_6$ x 33, $C_2^3:C_{12}$ x 27, $C_2^3.D_6$ x 27, $C_2^3:D_6$ x 24, $C_2^2.D_{12}$ x 18, $C_{12}:C_2^3$ x 12, $C_2^2\times D_{12}$ x 12, $C_{12}:C_2^3$ x 12, $C_2^3:C_{12}$ x 12, $C_2^3.D_6$ x 12, $C_6.C_2^4$ x 11, $D_4:D_6$ x 6, $D_{12}:C_2^2$ x 6, $Q_8:A_4$ x 4, $C_2^3\times A_4$ x 3, $C_2^4\times C_6$ x 2, $C_6.C_2^4$ x 2, $Q_8:D_6$ x 2, $Q_8\times D_6$ x 2, $C_2^2\times \SL(2,3)$

Order 72: $S_3\times D_6$ x 324, $C_6\times D_6$ x 116, $C_6:D_6$ x 58, $C_3:S_4$ x 48, $S_3\times A_4$ x 42, $C_3\times S_4$ x 42, $C_6\wr C_2$ x 36, $C_3:D_{12}$ x 36, $C_6\times A_4$ x 30, $S_3\times C_{12}$ x 20, $C_6.D_6$ x 20, $C_6^2:C_2$ x 18, $C_3\times D_{12}$ x 18, $D_6:S_3$ x 18, $C_6:C_{12}$ x 12, $C_{12}:S_3$ x 10, $C_6.D_6$ x 10, $C_2\times C_6^2$ x 9, $D_4\times C_3^2$ x 9, $C_3:D_{12}$ x 9, $C_6\times C_{12}$ x 6, $C_6.D_6$ x 6, $C_3\times \SL(2,3)$ x 6, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $Q_8\times C_3^2$, $C_3^2:Q_8$

Order 64: $C_2\wr C_2^2$ x 64, $C_2^3:D_4$ x 54, $C_2^4:C_4$ x 36, $D_4:C_2^3$ x 15, $D_4\times C_2^3$ x 6, $C_2^4:C_4$ x 6, $C_2^6$, $D_4:C_2^3$

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $C_2^2\times D_6$ x 458, $S_3\times D_4$ x 357, $C_6:D_4$ x 192, $C_2\times S_4$ x 123, $C_4\times D_6$ x 98, $C_6\times D_4$ x 96, $C_2\times D_{12}$ x 96, $D_6:C_4$ x 81, $D_4:S_3$ x 51, $D_{12}:C_2$ x 51, $C_2^2:C_{12}$ x 48, $C_6.D_4$ x 48, $C_2^3\times C_6$ x 33, $C_2^2\times A_4$ x 27, $D_4:C_6$ x 17, $D_{12}:C_2$ x 17, $C_2^2\times C_{12}$ x 12, $C_6.C_2^3$ x 12, $S_3\times Q_8$ x 9, $C_2\times \SL(2,3)$ x 5, $C_6\times Q_8$ x 2, $C_6:Q_8$ x 2

Order 36: $S_3^2$ x 123, $C_6\times S_3$ x 114, $C_6:S_3$ x 58, $C_6^2$ x 19, $C_3\times A_4$ x 18, $C_3:C_{12}$ x 8, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4

Order 32: $C_2^2\wr C_2$ x 120, $C_2^2\times D_4$ x 117, $C_2^3:C_4$ x 63, $D_4:C_2^2$ x 51, $C_2^3:C_4$ x 48, $C_2^5$ x 30, $D_4:C_2^2$ x 24, $C_2^3\times C_4$ x 6, $C_2^2\times Q_8$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 890, $C_3:D_4$ x 162, $C_2^2\times C_6$ x 140, $C_4\times S_3$ x 90, $D_{12}$ x 81, $C_3\times D_4$ x 81, $S_4$ x 54, $C_2\times C_{12}$ x 50, $C_6:C_4$ x 50, $C_2\times A_4$ x 42, $C_3:Q_8$ x 9, $C_3\times Q_8$ x 5, $\SL(2,3)$ x 4

Order 18: $C_3\times S_3$ x 42, $C_3:S_3$ x 23, $C_3\times C_6$ x 16

Order 16: $C_2\times D_4$ x 348, $C_2^4$ x 237, $C_2^2:C_4$ x 78, $D_4:C_2$ x 64, $C_2^2\times C_4$ x 58, $C_2\times Q_8$ x 6

Order 12: $D_6$ x 409, $C_2\times C_6$ x 144, $C_{12}$ x 20, $C_3:C_4$ x 20, $A_4$ x 12

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 474, $D_4$ x 144, $C_2\times C_4$ x 84, $Q_8$ x 6

Order 6: $S_3$ x 53, $C_6$ x 46

Order 4: $C_2^2$ x 222, $C_4$ x 16

Order 3: $C_3$ x 7

Order 2: $C_2$ x 27

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6912: $C_2^3:S_4\times S_3^2$

Order 3456: $C_6^2:C_2^2:S_4$, $Q_8:A_4:S_3^2$, $C_6^2:C_2^2:S_4$, $Q_8:A_4\times S_3^2$, $Q_8:A_4:S_3^2$

Order 2304: $C_2^4:D_6^2$, $S_3\times C_2^4:S_4$

Order 1728: $A_4:D_6^2$, $D_4:C_6^2:C_6$, $D_4:C_6^2:C_6$, $(C_2\times C_6^2):S_4$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$

Order 1152: $S_3\times C_2^3:S_4$ x 6, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $D_6^2:D_4$, $D_6^2:D_4$, $D_6^2:D_4$, $D_6^2:D_4$, $(C_6\times D_{12}):D_4$, $(C_2^2\times C_6^2):C_2^3$, $D_6^2.D_4$, $Q_8:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $D_4:D_6^2$, $D_6^2:D_4$

Order 864: $D_6^2:C_6$, $C_2\times A_4:S_3^2$, $S_3\times C_6:S_4$, $C_6:S_3\times S_4$, $C_6\times S_3\times S_4$, $S_4\times S_3^2$, $C_3^2\times Q_8:A_4$, $S_3^2\times \SL(2,3)$

Order 768: $C_2^5:S_4$, $C_2\wr C_2^2\times D_6$

Order 576: $(C_2^2\times C_6):S_4$ x 5, $S_3\times Q_8:A_4$ x 4, $C_3\times C_2^3:S_4$ x 4, $C_4:D_6^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_2^2$ x 2, $D_6^2:C_4$ x 2, $C_2^2\times D_6^2$, $C_2^2:D_6^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $C_6\times Q_8:A_4$, $D_6^2:C_2^2$, $D_6^2:C_2^2$, $C_4.D_6^2$, $D_6^2:C_2^2$, $D_6^2:C_2^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_2^4:S_3^2$, $C_2^4:C_6^2$, $(C_6\times C_{12}):D_4$, $(C_6\times C_{12}):D_4$, $(C_6\times D_4).D_6$, $(C_2\times C_{12}):D_{12}$, $(C_6\times C_{12}):D_4$, $C_3\times C_2^4:D_6$, $S_3\times C_2^3:C_{12}$, $C_3\times C_2^3:D_{12}$, $C_2^4:S_3^2$, $(C_2\times C_{12}):D_{12}$, $(C_2\times C_{12}):D_{12}$, $(C_6\times C_{12}):D_4$, $(C_6\times D_4).D_6$, $(C_6\times D_{12}):C_4$, $(C_2\times D_6):D_{12}$, $(C_2\times D_6):D_{12}$, $D_6^2:C_4$, $D_6^2:C_4$

Order 432: $C_6:S_3\times A_4$, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $C_6^2:D_6$, $A_4:C_6^2$, $C_2\times S_3^3$, $A_4\times S_3^2$, $A_4:S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_3:S_3\times \SL(2,3)$, $C_3\times S_3\times \SL(2,3)$

Order 384: $C_2\wr C_2^2\times S_3$ x 9, $C_2^4:S_4$ x 5, $D_{12}:C_2^4$, $C_2^5:A_4$, $C_2^5:D_6$, $C_2\wr C_2^2\times C_6$, $(C_6\times D_4):D_4$, $C_2^5:D_6$, $C_2^3:C_4\times D_6$, $C_2^4:D_{12}$

Order 288: $C_2\times D_6^2$ x 8, $D_6\times S_4$ x 7, $C_3\times Q_8:A_4$ x 4, $D_6^2:C_2$ x 4, $D_6^2:C_2$ x 4, $D_4\times S_3^2$ x 4, $D_6:D_{12}$ x 3, $D_6^2.C_2$ x 3, $D_6.D_{12}$ x 3, $C_6^2:C_2^3$ x 2, $C_2^3:S_3^2$ x 2, $D_{12}:D_6$ x 2, $D_6\times D_{12}$ x 2, $C_2.D_6^2$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$ x 2, $D_6^2:C_2$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $D_6:D_{12}$ x 2, $C_6^2:C_2^3$ x 2, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $D_{12}:D_6$, $D_{12}:D_6$, $D_{12}:D_6$, $Q_8\times S_3^2$, $D_{12}:D_6$, $D_{12}:D_6$, $D_4:S_3^2$, $D_{12}:D_6$, $D_{12}:D_6$, $D_{12}:D_6$, $D_{12}:D_6$, $D_{12}:D_6$, $D_6\times \SL(2,3)$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2:C_2^3$, $C_6^2:C_2^3$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $D_4:C_6^2$, $C_6^2.C_2^3$, $C_6^2.C_2^3$, $C_6^2.C_2^3$

Order 256: $C_2^5:D_4$

Order 216: $C_6:S_3^2$ x 2, $C_6\times S_3^2$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2\times S_4$, $S_3^3$, $C_3^2\times \SL(2,3)$

Order 192: $C_2^4:D_6$ x 13, $C_2^4:D_6$ x 7, $C_2^3:C_4\times S_3$ x 7, $C_2^3:D_{12}$ x 7, $D_{12}:C_2^3$ x 7, $C_2^3:S_4$ x 6, $C_3\times C_2\wr C_2^2$ x 5, $(C_2\times C_{12}):D_4$ x 5, $C_2^4:A_4$ x 3, $C_{12}:C_2^4$ x 2, $C_2^4:D_6$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_2^3.D_{12}$, $D_6\times C_2^4$, $C_2^3\times S_4$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $D_{12}:C_2^3$, $C_2^5:C_6$, $C_2^4:D_6$

Order 144: $D_6^2$ x 24, $C_6^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 6, $C_6:S_4$ x 6, $S_3\times S_4$ x 6, $A_4\times D_6$ x 5, $C_6\times S_4$ x 5, $D_6:D_6$ x 5, $D_6:D_6$ x 5, $C_6^2:C_2^2$ x 4, $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $C_4\times S_3^2$ x 4, $S_3\times \SL(2,3)$ x 4, $C_6.D_{12}$ x 3, $D_6:C_{12}$ x 3, $C_6.D_{12}$ x 3, $C_{12}:D_6$ x 3, $C_{12}:D_6$ x 3, $C_{12}:D_6$ x 3, $S_3\times D_{12}$ x 3, $C_6^2:C_4$ x 2, $C_6.D_{12}$ x 2, $C_4:C_6^2$ x 2, $C_6:D_{12}$ x 2, $C_{12}:D_6$ x 2, $C_6\times D_{12}$ x 2, $C_{12}\times D_6$ x 2, $D_6:D_6$ x 2, $C_2^2.S_3^2$ x 2, $D_6.D_6$ x 2, $C_6^2:C_4$ x 2, $C_6^2:C_4$ x 2, $C_2^2\times C_6^2$, $C_2^2:C_6^2$, $C_4.C_6^2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:C_6$, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:C_6$, $C_6\times \SL(2,3)$, $D_6.D_6$, $D_6.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $D_{12}:S_3$, $C_{12}.D_6$

Order 128: $C_2^4:D_4$ x 8, $D_4:C_2^4$, $C_2^4:D_4$, $C_2^5:C_4$

Order 108: $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3:S_3^2$ x 2, $C_3\times S_3^2$ x 2, $C_3^2:D_6$, $C_3^2\times A_4$, $C_3:S_3^2$

Order 96: $D_4\times D_6$ x 23, $C_2^2:D_{12}$ x 16, $D_6.D_4$ x 16, $D_6:D_4$ x 16, $C_2^3\times D_6$ x 13, $D_4:D_6$ x 9, $D_4:D_6$ x 8, $D_4:D_6$ x 7, $C_2^4:C_6$ x 7, $C_2^4:S_3$ x 7, $C_2^2\times S_4$ x 6, $C_2^2.D_{12}$ x 5, $C_2^3:C_{12}$ x 4, $C_2^3.D_6$ x 4, $C_6.C_2^4$ x 4, $C_2^3:D_6$ x 4, $Q_8:A_4$ x 3, $C_2^2.D_{12}$ x 3, $C_{12}:C_2^3$ x 2, $C_2^2\times D_{12}$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3:C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^4\times C_6$, $C_2^3\times A_4$, $C_6.C_2^4$, $Q_8:D_6$, $Q_8\times D_6$, $D_4:D_6$, $D_{12}:C_2^2$, $C_2^2\times \SL(2,3)$

Order 72: $S_3\times D_6$ x 32, $C_6\times D_6$ x 15, $C_6:D_6$ x 14, $C_6\times A_4$ x 5, $C_3:S_4$ x 5, $C_2\times C_6^2$ x 4, $S_3\times A_4$ x 4, $C_3\times S_4$ x 4, $C_3\times \SL(2,3)$ x 4, $C_6^2:C_2$ x 3, $C_{12}:S_3$ x 3, $C_6\wr C_2$ x 3, $S_3\times C_{12}$ x 3, $C_3:D_{12}$ x 3, $C_6.D_6$ x 3, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:D_{12}$ x 2, $C_6:C_{12}$ x 2, $C_3\times D_{12}$ x 2, $D_6:S_3$ x 2, $Q_8\times C_3^2$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3^2:Q_8$

Order 64: $C_2\wr C_2^2$ x 13, $C_2^3:D_4$ x 12, $C_2^4:C_4$ x 6, $D_4:C_2^3$ x 6, $D_4\times C_2^3$ x 2, $C_2^4:C_4$ x 2, $C_2^6$, $D_4:C_2^3$

Order 54: $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2\times C_6$, $C_3^2:S_3$

Order 48: $C_2^2\times D_6$ x 50, $C_6:D_4$ x 25, $S_3\times D_4$ x 25, $C_4\times D_6$ x 15, $D_6:C_4$ x 15, $C_6\times D_4$ x 13, $C_2\times D_{12}$ x 13, $C_2\times S_4$ x 11, $C_2^2:C_{12}$ x 9, $C_6.D_4$ x 9, $C_2^3\times C_6$ x 8, $D_4:S_3$ x 7, $D_{12}:C_2$ x 7, $D_4:C_6$ x 5, $D_{12}:C_2$ x 5, $S_3\times Q_8$ x 5, $C_2^2\times A_4$ x 4, $C_2\times \SL(2,3)$ x 3, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2, $C_6\times Q_8$, $C_6:Q_8$

Order 36: $C_6\times S_3$ x 20, $C_6:S_3$ x 16, $S_3^2$ x 15, $C_6^2$ x 7, $C_3\times A_4$ x 4, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2

Order 32: $C_2^2\wr C_2$ x 24, $C_2^2\times D_4$ x 20, $C_2^3:C_4$ x 14, $D_4:C_2^2$ x 12, $C_2^5$ x 10, $D_4:C_2^2$ x 8, $C_2^3:C_4$ x 7, $C_2^3\times C_4$ x 2, $C_2^2\times Q_8$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 81, $C_2^2\times C_6$ x 22, $C_3:D_4$ x 15, $C_4\times S_3$ x 15, $C_2\times C_{12}$ x 9, $C_6:C_4$ x 9, $D_{12}$ x 9, $C_3\times D_4$ x 9, $C_2\times A_4$ x 6, $S_4$ x 6, $C_3:Q_8$ x 4, $\SL(2,3)$ x 3, $C_3\times Q_8$ x 3

Order 18: $C_3\times S_3$ x 11, $C_3\times C_6$ x 8, $C_3:S_3$ x 8

Order 16: $C_2\times D_4$ x 42, $C_2^4$ x 36, $C_2^2:C_4$ x 17, $D_4:C_2$ x 13, $C_2^2\times C_4$ x 12, $C_2\times Q_8$ x 4

Order 12: $D_6$ x 53, $C_2\times C_6$ x 26, $C_{12}$ x 6, $C_3:C_4$ x 6, $A_4$ x 3

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 55, $C_2\times C_4$ x 18, $D_4$ x 17, $Q_8$ x 4

Order 6: $C_6$ x 15, $S_3$ x 15

Order 4: $C_2^2$ x 35, $C_4$ x 6

Order 3: $C_3$ x 5

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6912: $C_2^3:S_4\times S_3^2$ ($C_1$)

Order 3456: $Q_8:A_4:S_3^2$ ($C_2$) x 2, $C_6^2:C_2^2:S_4$ ($C_2$) x 2, $C_6^2:C_2^2:S_4$ ($C_2$), $Q_8:A_4\times S_3^2$ ($C_2$), $Q_8:A_4:S_3^2$ ($C_2$)

Order 1728: $D_4:C_6^2:C_6$ ($C_2^2$) x 2, $(C_2\times C_6^2):S_4$ ($C_2^2$) x 2, $D_4:C_6^2:C_6$ ($C_2^2$), $(C_2\times C_6^2):S_4$ ($C_2^2$), $C_3^2\times C_2^3:S_4$ ($C_2^2$)

Order 1152: $S_3\times C_2^3:S_4$ ($S_3$) x 2, $D_4:D_6^2$ ($S_3$)

Order 864: $C_3^2\times Q_8:A_4$ ($C_2^3$)

Order 576: $C_6^2.C_2^4$ ($D_6$) x 2, $S_3\times Q_8:A_4$ ($D_6$) x 2, $(C_2^2\times C_6):S_4$ ($D_6$) x 2, $C_3\times C_2^3:S_4$ ($D_6$) x 2, $C_6^2.C_2^4$ ($D_6$)

Order 288: $C_2\times D_6^2$ ($S_4$) x 3, $C_3\times Q_8:A_4$ ($C_2\times D_6$) x 2, $D_4:C_6^2$ ($C_2\times D_6$)

Order 192: $D_{12}:C_2^3$ ($S_3^2$) x 2, $C_2^3:S_4$ ($S_3^2$)

Order 144: $C_6^2:C_2^2$ ($C_2\times S_4$) x 6, $C_6^2:C_2^2$ ($C_2\times S_4$) x 3

Order 96: $C_6.C_2^4$ ($S_3\times D_6$) x 2, $Q_8:A_4$ ($S_3\times D_6$)

Order 72: $C_2\times C_6^2$ ($C_2^2\times S_4$) x 3, $S_3\times D_6$ ($C_2^2:S_4$)

Order 48: $C_2^2\times D_6$ ($S_3\times S_4$) x 6

Order 36: $S_3^2$ ($C_2^3:S_4$) x 4, $C_6\times S_3$ ($C_2^3:S_4$) x 2, $C_6:S_3$ ($C_2^3:S_4$)

Order 32: $D_4:C_2^2$ ($S_3^3$)

Order 24: $C_2^2\times C_6$ ($D_6\times S_4$) x 6

Order 18: $C_3\times S_3$ ($C_2^4:S_4$) x 4, $C_3:S_3$ ($C_2^4:S_4$) x 2, $C_3\times C_6$ ($C_2^2\wr S_3$)

Order 12: $D_6$ ($C_2^4:S_3^2$) x 2

Order 9: $C_3^2$ ($C_2^5:S_4$)

Order 8: $C_2^3$ ($S_4\times S_3^2$) x 3

Order 6: $S_3$ ($S_3\times C_2^3:S_4$) x 4, $C_6$ ($C_2^5:S_3^2$) x 2

Order 3: $C_3$ ($S_3\times C_2^4:S_4$) x 2

Order 2: $C_2$ ($D_6^2:S_4$)

Order 1: $C_1$ ($C_2^3:S_4\times S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6912: $C_2^3:S_4\times S_3^2$ ($C_1$)

Order 3456: $C_6^2:C_2^2:S_4$ ($C_2$), $Q_8:A_4:S_3^2$ ($C_2$), $C_6^2:C_2^2:S_4$ ($C_2$), $Q_8:A_4\times S_3^2$ ($C_2$), $Q_8:A_4:S_3^2$ ($C_2$)

Order 1728: $D_4:C_6^2:C_6$ ($C_2^2$), $D_4:C_6^2:C_6$ ($C_2^2$), $(C_2\times C_6^2):S_4$ ($C_2^2$), $(C_2\times C_6^2):S_4$ ($C_2^2$), $C_3^2\times C_2^3:S_4$ ($C_2^2$)

Order 1152: $S_3\times C_2^3:S_4$ ($S_3$), $D_4:D_6^2$ ($S_3$)

Order 864: $C_3^2\times Q_8:A_4$ ($C_2^3$)

Order 576: $C_6^2.C_2^4$ ($D_6$), $C_6^2.C_2^4$ ($D_6$), $S_3\times Q_8:A_4$ ($D_6$), $(C_2^2\times C_6):S_4$ ($D_6$), $C_3\times C_2^3:S_4$ ($D_6$)

Order 288: $C_3\times Q_8:A_4$ ($C_2\times D_6$), $C_2\times D_6^2$ ($S_4$), $D_4:C_6^2$ ($C_2\times D_6$)

Order 192: $D_{12}:C_2^3$ ($S_3^2$), $C_2^3:S_4$ ($S_3^2$)

Order 144: $C_6^2:C_2^2$ ($C_2\times S_4$), $C_6^2:C_2^2$ ($C_2\times S_4$)

Order 96: $C_6.C_2^4$ ($S_3\times D_6$), $Q_8:A_4$ ($S_3\times D_6$)

Order 72: $C_2\times C_6^2$ ($C_2^2\times S_4$), $S_3\times D_6$ ($C_2^2:S_4$)

Order 48: $C_2^2\times D_6$ ($S_3\times S_4$)

Order 36: $C_6:S_3$ ($C_2^3:S_4$), $C_6\times S_3$ ($C_2^3:S_4$), $S_3^2$ ($C_2^3:S_4$)

Order 32: $D_4:C_2^2$ ($S_3^3$)

Order 24: $C_2^2\times C_6$ ($D_6\times S_4$)

Order 18: $C_3\times C_6$ ($C_2^2\wr S_3$), $C_3:S_3$ ($C_2^4:S_4$), $C_3\times S_3$ ($C_2^4:S_4$)

Order 12: $D_6$ ($C_2^4:S_3^2$)

Order 9: $C_3^2$ ($C_2^5:S_4$)

Order 8: $C_2^3$ ($S_4\times S_3^2$)

Order 6: $C_6$ ($C_2^5:S_3^2$), $S_3$ ($S_3\times C_2^3:S_4$)

Order 3: $C_3$ ($S_3\times C_2^4:S_4$)

Order 2: $C_2$ ($D_6^2:S_4$)

Order 1: $C_1$ ($C_2^3:S_4\times S_3^2$)

Series

Derived series

$C_2^3:S_4\times S_3^2$

$\rhd$

$C_2^3:S_4\times S_3^2$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_2^3:S_4\times S_3^2$

$\rhd$

$C_2^3:S_4\times S_3^2$

$\rhd$

$C_6^2:C_2^2:S_4$

$\rhd$

$C_6^2:C_2^2:S_4$

$\rhd$

$D_4:C_6^2:C_6$

$\rhd$

$D_4:C_6^2:C_6$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3\times Q_8:A_4$

$\rhd$

$D_4:C_6^2$

$\rhd$

$D_4:C_6^2$

$\rhd$

$C_6.C_2^4$

$\rhd$

$C_6.C_2^4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^3:S_4\times S_3^2$

$\rhd$

$C_2^3:S_4\times S_3^2$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3^2\times Q_8:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 2 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 $117 \times 117$ rational character table. Alternatively, you may search for characters of this group with desired properties.