Abstract group 3456.qg: $C_6.S_4^2$

• Label: 3456.qg
• Order: \( 2^{7} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $108$
• Rank: $2$

Group information

Description:	$C_6.S_4^2$
Order:	\(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.S_3^3$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Composition factors:	$C_2$ x 7, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	247	242	648	1022	1296	3456
Conjugacy classes	1	10	8	9	26	12	66
Divisions	1	10	7	9	23	6	56
Autjugacy classes	1	10	7	9	23	6	56

Dimension	1	2	3	4	6	8	9	12	18	36
Irr. complex chars.	4	13	8	10	18	0	4	4	5	0	66
Irr. rational chars.	4	7	8	9	10	2	4	8	3	1	56

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$108$
Rank:	$2$
Inequivalent generating pairs:	$54$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{12}=c^{2}=d^{6}=e^{2}=f^{6}=[a,c]=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3,2)(4,5,7,6)(8,14)(11,13)(12,15), (4,5,6)(10,13,11), (1,2,3)(4,6,7) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.S_4)$ $\,\rtimes\,$ $S_4$	$(C_6:S_4)$ $\,\rtimes\,$ $S_4$	$A_4$ $\,\rtimes\,$ $(D_6:S_4)$	$(A_4^2:D_6)$ $\,\rtimes\,$ $C_2$	all 25
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $S_4^2$	$C_2$ . $(C_3:S_4^2)$	$(C_2\times A_4^2)$ . $D_6$	$(C_2^3\times A_4)$ . $S_3^2$ (2)	all 12

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

Homology

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

Subgroups

There are 29096 subgroups in 1290 conjugacy classes, 45 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3:S_4^2$
Commutator:	$G' \simeq$ $C_6\times A_4^2$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3:S_4^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_6$	$G/\operatorname{Fit} \simeq$ $S_3^2$
Radical:	$R \simeq$ $C_6.S_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_6$	$G/\operatorname{soc} \simeq$ $S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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 3456: $C_6.S_4^2$

Order 1728: $C_6\times \PSOPlus(4,3)$, $A_4^2:D_6$, $A_4\times C_6.S_4$

Order 1152: $A_4:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$

Order 864: $C_6\times A_4^2$, $A_4^2:C_6$, $A_4^2:S_3$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$

Order 576: $A_4^2:C_2^2$ x 2, $C_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $A_4^2:C_4$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:S_4\times C_6$, $C_2^2\times C_6:S_4$, $A_4^2:C_2^2$, $C_6:D_4\times A_4$, $C_2^2\times C_6.S_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $A_4\times C_6.D_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$

Order 432: $C_6^2:D_6$ x 3, $C_6:S_3\times A_4$, $C_3\times A_4^2$, $C_6^2:C_{12}$, $C_3\times C_6.S_4$

Order 384: $C_2^2:\GL(2,\mathbb{Z}/4)$, $C_2^4:D_{12}$, $C_6.D_4^2$

Order 288: $C_2^3.S_3^2$ x 5, $C_3\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2\times C_6:S_4$ x 4, $C_2\times A_4^2$ x 4, $C_6:C_4\times A_4$ x 3, $A_4\times S_4$ x 3, $A_4\times C_3:D_4$ x 2, $C_2\times C_6.S_4$ x 2, $C_2\times A_4:C_{12}$ x 2, $D_6:S_4$ x 2, $C_2^3:C_6^2$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $D_6:S_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_2^5:C_3^2$, $C_2\times A_4\times D_6$, $\PSOPlus(4,3)$

Order 216: $C_3^2:S_4$ x 3, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:D_{12}$

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3\times S_4$ x 2, $C_2^3.S_4$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^4.D_6$, $C_2^3:S_4$, $D_6:C_2^4$, $C_2^5:C_6$, $C_2^5:C_6$, $C_2^4.D_6$, $C_2^4:D_6$

Order 144: $C_6\times S_4$ x 11, $C_6:S_4$ x 9, $C_2^2:C_6^2$ x 6, $A_4\times D_6$ x 5, $A_4\times C_3:C_4$ x 5, $A_4^2$ x 4, $A_4:C_{12}$ x 4, $C_6^2:C_2^2$ x 3, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_6:D_{12}$ x 2, $C_6.S_4$ x 2, $C_6^2:C_4$, $C_2^2.S_3^2$, $C_6^2:C_2^2$, $C_2^4:C_3^2$, $C_6^2:C_4$, $D_6:D_6$

Order 128: $C_2^4:D_4$

Order 108: $C_3^2:D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3^2:C_{12}$

Order 96: $C_2^2\times S_4$ x 12, $C_2^3:D_6$ x 10, $C_2^3.D_6$ x 9, $\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2.S_4$ x 6, $C_2^3.D_6$ x 6, $C_2^2.D_{12}$ x 6, $C_2^3\times A_4$ x 5, $C_6.C_4^2$ x 4, $C_4:S_4$ x 4, $C_2^4:C_6$ x 4, $D_6:D_4$ x 4, $C_{12}:C_2^3$ x 3, $C_6.C_2^4$ x 3, $C_2^3:C_{12}$ x 3, $C_2^3:C_{12}$ x 3, $C_2^3:A_4$ x 2, $C_2^2:S_4$ x 2, $D_4\times A_4$ x 2, $C_2^3.D_6$ x 2, $C_2^2:D_{12}$, $C_2^4\times C_6$, $C_2^3\times D_6$

Order 72: $C_6\times A_4$ x 17, $C_3\times S_4$ x 11, $C_6\wr C_2$ x 6, $C_6:C_{12}$ x 5, $C_3:D_{12}$ x 5, $C_6:D_6$ x 4, $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_6\times D_6$ x 3, $C_2\times C_6^2$ x 2, $C_6.D_6$ x 2, $D_6:S_3$ x 2

Order 64: $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^3.D_4$, $C_2^4:C_4$, $D_4\times C_2^3$, $C_2^4:C_4$

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

Order 48: $C_6:D_4$ x 36, $C_2\times S_4$ x 26, $C_6\times D_4$ x 17, $C_2^2\times A_4$ x 15, $C_6.C_2^3$ x 14, $D_6:C_4$ x 13, $C_2^2\times D_6$ x 10, $C_6.D_4$ x 10, $C_2^3\times C_6$ x 8, $C_2^2:C_{12}$ x 7, $A_4:C_4$ x 6, $C_6.D_4$ x 5, $C_2^2\times C_{12}$ x 3, $C_4\times A_4$ x 3, $C_2^2:A_4$ x 2, $C_2\times D_{12}$ x 2

Order 36: $C_3\times A_4$ x 13, $C_6\times S_3$ x 12, $C_6:S_3$ x 8, $C_3:C_{12}$ x 7, $C_6^2$ x 6, $C_3^2:C_4$ x 2

Order 32: $C_2^3:C_4$ x 15, $C_2^2\times D_4$ x 13, $C_4:D_4$ x 8, $C_2^2\wr C_2$ x 8, $C_2.C_4^2$ x 4, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_2^2.D_4$ x 2

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 33, $C_2^2\times C_6$ x 31, $C_3:D_4$ x 26, $C_6:C_4$ x 25, $C_2\times A_4$ x 19, $S_4$ x 14, $C_3\times D_4$ x 14, $C_2\times C_{12}$ x 12, $D_{12}$ x 4

Order 18: $C_3\times C_6$ x 11, $C_3\times S_3$ x 8, $C_3:S_3$ x 3

Order 16: $C_2\times D_4$ x 49, $C_2^2:C_4$ x 25, $C_2^4$ x 16, $C_2^2\times C_4$ x 15, $C_4:C_4$ x 4

Order 12: $C_2\times C_6$ x 41, $D_6$ x 40, $C_3:C_4$ x 12, $A_4$ x 10, $C_{12}$ x 6

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 47, $D_4$ x 32, $C_2\times C_4$ x 28

Order 6: $C_6$ x 23, $S_3$ x 12

Order 4: $C_2^2$ x 42, $C_4$ x 9

Order 3: $C_3$ x 7

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3456: $C_6.S_4^2$

Order 1728: $C_6\times \PSOPlus(4,3)$, $A_4^2:D_6$, $A_4\times C_6.S_4$

Order 1152: $A_4:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$

Order 864: $C_6\times A_4^2$, $A_4^2:C_6$, $A_4^2:S_3$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$

Order 576: $A_4^2:C_2^2$ x 2, $C_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $A_4^2:C_4$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:S_4\times C_6$, $C_2^2\times C_6:S_4$, $A_4^2:C_2^2$, $C_6:D_4\times A_4$, $C_2^2\times C_6.S_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $A_4\times C_6.D_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$

Order 432: $C_6^2:D_6$ x 3, $C_6:S_3\times A_4$, $C_3\times A_4^2$, $C_6^2:C_{12}$, $C_3\times C_6.S_4$

Order 384: $C_2^2:\GL(2,\mathbb{Z}/4)$, $C_2^4:D_{12}$, $C_6.D_4^2$

Order 288: $C_3\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^3.S_3^2$ x 4, $C_2\times C_6:S_4$ x 4, $C_2\times A_4^2$ x 4, $C_6:C_4\times A_4$ x 3, $A_4\times C_3:D_4$ x 2, $C_2\times C_6.S_4$ x 2, $C_2\times A_4:C_{12}$ x 2, $D_6:S_4$ x 2, $C_2^3:C_6^2$ x 2, $C_2\times C_6\times S_4$ x 2, $A_4\times S_4$ x 2, $D_6:S_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_2^5:C_3^2$, $C_2\times A_4\times D_6$, $C_3\times C_2^2:S_4$, $\PSOPlus(4,3)$

Order 216: $C_3^2:S_4$ x 3, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:D_{12}$

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2\times \GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3\times S_4$ x 2, $C_2^3.S_4$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4:C_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_6$, $C_2^3.D_{12}$, $C_2^4.D_6$, $C_2^3:S_4$, $D_6:C_2^4$, $C_2^5:C_6$, $C_2^5:C_6$, $C_2^4.D_6$, $C_2^4:D_6$

Order 144: $C_6\times S_4$ x 11, $C_6:S_4$ x 8, $C_2^2:C_6^2$ x 6, $A_4\times D_6$ x 5, $A_4\times C_3:C_4$ x 5, $A_4^2$ x 4, $A_4:C_{12}$ x 4, $C_6^2:C_2^2$ x 3, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_6:D_{12}$ x 2, $C_6.S_4$ x 2, $C_6^2:C_4$, $C_2^2.S_3^2$, $C_6^2:C_2^2$, $C_2^4:C_3^2$, $C_6^2:C_4$, $D_6:D_6$

Order 128: $C_2^4:D_4$

Order 108: $C_3^2:D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3^2:C_{12}$

Order 96: $C_2^2\times S_4$ x 10, $C_2^3:D_6$ x 10, $C_2^3.D_6$ x 7, $\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2.S_4$ x 6, $C_2^3.D_6$ x 6, $C_2^2.D_{12}$ x 6, $C_2^3\times A_4$ x 5, $C_6.C_4^2$ x 4, $C_2^4:C_6$ x 4, $D_6:D_4$ x 4, $C_{12}:C_2^3$ x 3, $C_6.C_2^4$ x 3, $C_2^3:C_{12}$ x 3, $C_4:S_4$ x 3, $C_2^3:C_{12}$ x 3, $C_2^3:A_4$ x 2, $D_4\times A_4$ x 2, $C_2^3.D_6$ x 2, $C_2^2:D_{12}$, $C_2^4\times C_6$, $C_2^3\times D_6$, $C_2^2:S_4$

Order 72: $C_6\times A_4$ x 17, $C_3\times S_4$ x 7, $C_6\wr C_2$ x 6, $C_6:C_{12}$ x 5, $C_6:D_6$ x 4, $C_3:S_4$ x 4, $C_3:D_{12}$ x 4, $C_6\times D_6$ x 3, $S_3\times A_4$ x 3, $C_2\times C_6^2$ x 2, $C_6.D_6$ x 2, $D_6:S_3$ x 2

Order 64: $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^3.D_4$, $C_2^4:C_4$, $D_4\times C_2^3$, $C_2^4:C_4$

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

Order 48: $C_6:D_4$ x 31, $C_2\times S_4$ x 21, $C_6\times D_4$ x 17, $C_2^2\times A_4$ x 15, $C_6.C_2^3$ x 14, $D_6:C_4$ x 12, $C_2^2\times D_6$ x 10, $C_6.D_4$ x 9, $C_2^3\times C_6$ x 8, $C_2^2:C_{12}$ x 7, $A_4:C_4$ x 6, $C_6.D_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_4\times A_4$ x 3, $C_2^2:A_4$ x 2, $C_2\times D_{12}$ x 2

Order 36: $C_3\times A_4$ x 13, $C_6\times S_3$ x 12, $C_3:C_{12}$ x 7, $C_6:S_3$ x 7, $C_6^2$ x 6, $C_3^2:C_4$ x 2

Order 32: $C_2^3:C_4$ x 15, $C_2^2\times D_4$ x 13, $C_2^2\wr C_2$ x 8, $C_4:D_4$ x 6, $C_2.C_4^2$ x 4, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_2^2.D_4$ x 2

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 31, $C_2\times D_6$ x 29, $C_6:C_4$ x 24, $C_3:D_4$ x 22, $C_2\times A_4$ x 19, $C_2\times C_{12}$ x 12, $C_3\times D_4$ x 12, $S_4$ x 9, $D_{12}$ x 3

Order 18: $C_3\times C_6$ x 11, $C_3\times S_3$ x 6, $C_3:S_3$ x 3

Order 16: $C_2\times D_4$ x 44, $C_2^2:C_4$ x 23, $C_2^4$ x 16, $C_2^2\times C_4$ x 15, $C_4:C_4$ x 3

Order 12: $C_2\times C_6$ x 41, $D_6$ x 33, $C_3:C_4$ x 12, $A_4$ x 10, $C_{12}$ x 6

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 45, $C_2\times C_4$ x 27, $D_4$ x 26

Order 6: $C_6$ x 23, $S_3$ x 9

Order 4: $C_2^2$ x 40, $C_4$ x 9

Order 3: $C_3$ x 7

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3456: $C_6.S_4^2$ ($C_1$)

Order 1728: $C_6\times \PSOPlus(4,3)$ ($C_2$), $A_4^2:D_6$ ($C_2$), $A_4\times C_6.S_4$ ($C_2$)

Order 864: $C_6\times A_4^2$ ($C_2^2$)

Order 576: $C_2^2\times C_6:S_4$ ($S_3$), $A_4^2:C_2^2$ ($S_3$), $C_2^2\times C_6.S_4$ ($S_3$)

Order 432: $C_3\times A_4^2$ ($D_4$)

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

Order 144: $C_2^2:C_6^2$ ($C_3:D_4$), $C_2^2:C_6^2$ ($D_{12}$), $C_6:S_4$ ($S_4$), $A_4^2$ ($C_3:D_4$), $C_6.S_4$ ($S_4$)

Order 96: $C_2^3\times A_4$ ($S_3^2$) x 2, $C_2^4\times C_6$ ($S_3^2$)

Order 72: $C_6\times A_4$ ($C_2\times S_4$) x 2

Order 48: $C_2^3\times C_6$ ($C_3:D_{12}$), $C_2^2\times A_4$ ($C_3:D_{12}$), $C_2^2\times A_4$ ($D_6:S_3$)

Order 36: $C_3\times A_4$ ($\GL(2,\mathbb{Z}/4)$), $C_3\times A_4$ ($C_4:S_4$)

Order 32: $C_2^5$ ($C_3:S_3^2$)

Order 24: $C_2^2\times C_6$ ($S_3\times S_4$) x 2, $C_2\times A_4$ ($S_3\times S_4$) x 2

Order 16: $C_2^4$ ($C_3^2:D_{12}$)

Order 12: $C_2\times C_6$ ($D_6:S_4$), $C_2\times C_6$ ($C_2^3.S_3^2$), $A_4$ ($D_6:S_4$), $A_4$ ($C_2^3.S_3^2$)

Order 8: $C_2^3$ ($A_4:S_3^2$) x 2

Order 6: $C_6$ ($S_4^2$)

Order 4: $C_2^2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$), $C_2^2$ ($C_6^2:D_{12}$)

Order 3: $C_3$ ($A_4:\GL(2,\mathbb{Z}/4)$)

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

Order 1: $C_1$ ($C_6.S_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3456: $C_6.S_4^2$ ($C_1$)

Order 1728: $C_6\times \PSOPlus(4,3)$ ($C_2$), $A_4^2:D_6$ ($C_2$), $A_4\times C_6.S_4$ ($C_2$)

Order 864: $C_6\times A_4^2$ ($C_2^2$)

Order 576: $C_2^2\times C_6:S_4$ ($S_3$), $A_4^2:C_2^2$ ($S_3$), $C_2^2\times C_6.S_4$ ($S_3$)

Order 432: $C_3\times A_4^2$ ($D_4$)

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

Order 144: $C_2^2:C_6^2$ ($C_3:D_4$), $C_2^2:C_6^2$ ($D_{12}$), $C_6:S_4$ ($S_4$), $A_4^2$ ($C_3:D_4$), $C_6.S_4$ ($S_4$)

Order 96: $C_2^3\times A_4$ ($S_3^2$) x 2, $C_2^4\times C_6$ ($S_3^2$)

Order 72: $C_6\times A_4$ ($C_2\times S_4$) x 2

Order 48: $C_2^3\times C_6$ ($C_3:D_{12}$), $C_2^2\times A_4$ ($C_3:D_{12}$), $C_2^2\times A_4$ ($D_6:S_3$)

Order 36: $C_3\times A_4$ ($\GL(2,\mathbb{Z}/4)$), $C_3\times A_4$ ($C_4:S_4$)

Order 32: $C_2^5$ ($C_3:S_3^2$)

Order 24: $C_2^2\times C_6$ ($S_3\times S_4$) x 2, $C_2\times A_4$ ($S_3\times S_4$) x 2

Order 16: $C_2^4$ ($C_3^2:D_{12}$)

Order 12: $C_2\times C_6$ ($D_6:S_4$), $C_2\times C_6$ ($C_2^3.S_3^2$), $A_4$ ($D_6:S_4$), $A_4$ ($C_2^3.S_3^2$)

Order 8: $C_2^3$ ($A_4:S_3^2$) x 2

Order 6: $C_6$ ($S_4^2$)

Order 4: $C_2^2$ ($C_3^2:\GL(2,\mathbb{Z}/4)$), $C_2^2$ ($C_6^2:D_{12}$)

Order 3: $C_3$ ($A_4:\GL(2,\mathbb{Z}/4)$)

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

Order 1: $C_1$ ($C_6.S_4^2$)

Series

Derived series

$C_6.S_4^2$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_6.S_4^2$

$\rhd$

$A_4^2:D_6$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6.S_4^2$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_3\times A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 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 $66 \times 66$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $56 \times 56$ rational character table.