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

• Label: 3456.jm
• Order: \( 2^{7} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $13$
• Trans deg.: $54$
• Rank: $3$

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:	$S_3\times S_4^2:D_4$, of order \(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Composition factors:	$C_2$ x 7, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	391	242	504	1310	1008	3456
Conjugacy classes	1	17	8	14	28	10	78
Divisions	1	17	7	14	27	10	76
Autjugacy classes	1	8	5	4	15	3	36

Dimension	1	2	3	4	6	8	9	12	18
Irr. complex chars.	8	12	16	10	16	0	8	4	4	78
Irr. rational chars.	8	12	16	6	16	2	8	4	4	76

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$302400$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{6}=d^{2}=e^{2}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(8,9)(10,13), (1,2)(3,4)(6,7)(8,10,9,13)(11,12), (2,3,4)(5,6,7)(8,11,13,9,12,10) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3:S_4^2)$
Semidirect product:	$C_6$ $\,\rtimes\,$ $S_4^2$	$(C_6:S_4)$ $\,\rtimes\,$ $S_4$	$A_4$ $\,\rtimes\,$ $(D_6\times S_4)$	$C_3$ $\,\rtimes\,$ $(C_2\times S_4^2)$	all 32
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut((C_2\times C_6).D_6^2)$

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 63964 subgroups in 2916 conjugacy classes, 65 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3:S_4^2$
Commutator:	$G' \simeq$ $C_3\times A_4^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_6:S_4^2$
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:	$P_{ 2 } \simeq$ $C_2\times D_4^2$
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: $C_3:S_4^2$ x 4, $A_4^2:D_6$ x 2, $C_6\times \PSOPlus(4,3)$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 2, $C_2\times S_4^2$

Order 864: $A_4^2:S_3$ x 4, $C_2\times A_4:S_3^2$ x 2, $A_4^2:C_6$ x 2, $C_6\times A_4^2$

Order 576: $\GL(2,\mathbb{Z}/4):S_3$ x 16, $A_4^2:C_2^2$ x 4, $S_4^2$ x 4, $C_2^2\times C_6:S_4$ x 2, $C_2^2:D_6^2$ x 2, $C_6:D_4\times A_4$ x 2, $C_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $C_2^4.S_3^2$ x 2, $C_2^2:S_4\times C_6$, $A_4^2:C_2^2$

Order 432: $A_4:S_3^2$ x 8, $C_6^2:D_6$ x 4, $C_6:S_3\times A_4$ x 2, $C_3\times A_4^2$

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

Order 288: $D_6\times S_4$ x 18, $C_2\times C_6:S_4$ x 12, $A_4\times C_3:D_4$ x 8, $C_3\times \GL(2,\mathbb{Z}/4)$ x 8, $D_6:S_4$ x 8, $C_2^3.S_3^2$ x 8, $C_2^3.S_3^2$ x 8, $A_4\times S_4$ x 8, $C_2\times A_4^2$ x 4, $D_6^2:C_2$ x 2, $C_6:C_4\times A_4$ x 2, $C_2\times A_4:C_{12}$ x 2, $C_2^3:C_6^2$ x 2, $C_2\times A_4\times D_6$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $\PSOPlus(4,3)$ x 2, $C_2^5:C_3^2$

Order 216: $C_3^2:S_4$ x 8, $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 2, $C_6:S_3^2$

Order 192: $D_4\times S_4$ x 16, $C_2^4:D_6$ x 16, $C_2^3\times S_4$ x 6, $C_2\times \GL(2,\mathbb{Z}/4)$ x 4, $D_6:C_2^4$ x 2, $C_{12}:C_2^4$ x 2, $C_2^5:C_6$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^3:S_4$, $C_2^5:C_6$, $C_2^4.D_6$, $C_2^4:D_6$

Order 144: $S_3\times S_4$ x 24, $C_6:S_4$ x 20, $D_6:D_6$ x 16, $C_6\times S_4$ x 15, $A_4\times D_6$ x 12, $C_2^2:C_6^2$ x 6, $A_4^2$ x 4, $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $A_4\times C_3:C_4$ x 4, $A_4:C_{12}$ x 4, $C_6^2:C_2^2$ x 2, $D_6^2$ x 2, $C_2^2.S_3^2$ x 2, $C_2^4:C_3^2$

Order 128: $C_2\times D_4^2$

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

Order 96: $C_2^2\times S_4$ x 42, $D_4\times D_6$ x 34, $C_2^3:D_6$ x 26, $D_6:D_4$ x 16, $C_2^2:D_{12}$ x 16, $C_2^3.D_6$ x 16, $\GL(2,\mathbb{Z}/4)$ x 16, $C_2^3.D_6$ x 16, $D_4\times A_4$ x 8, $C_4:S_4$ x 8, $C_4\times S_4$ x 8, $C_2^4:C_6$ x 8, $C_{12}:D_4$ x 8, $D_6:D_4$ x 8, $C_2^3\times A_4$ x 6, $C_2^2.D_{12}$ x 4, $C_2^3\times D_6$ x 3, $C_{12}:C_2^3$ x 3, $C_2^2\times D_{12}$ x 3, $C_2^3:C_{12}$ x 3, $C_2^3:A_4$ x 2, $C_2^2:S_4$ x 2, $C_6.C_2^4$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3:C_{12}$ x 2, $C_2^2.S_4$ x 2, $C_2^3.D_6$ x 2, $C_2^4\times C_6$, $C_2^3.D_6$, $C_6:C_4^2$

Order 72: $C_3\times S_4$ x 18, $C_6\times A_4$ x 17, $S_3\times D_6$ x 17, $C_6\wr C_2$ x 16, $C_3:D_{12}$ x 16, $C_6:D_6$ x 12, $S_3\times A_4$ x 12, $C_3:S_4$ x 12, $C_6.D_6$ x 8, $C_6\times D_6$ x 4, $C_6:C_{12}$ x 4, $C_2\times C_6^2$ x 2

Order 64: $D_4^2$ x 16, $D_4\times C_2^3$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$

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

Order 48: $C_6:D_4$ x 82, $S_3\times D_4$ x 80, $C_2\times S_4$ x 65, $C_2^2\times D_6$ x 40, $C_6\times D_4$ x 24, $C_2\times D_{12}$ x 24, $C_2^2\times A_4$ x 20, $C_4\times D_6$ x 18, $D_6:C_4$ x 16, $C_6.C_2^3$ x 12, $C_2^2:C_{12}$ x 12, $C_2^3\times C_6$ x 8, $C_6.D_4$ x 8, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_6.D_4$ x 4, $C_{12}:C_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_2^2:A_4$ x 2

Order 36: $C_6\times S_3$ x 21, $S_3^2$ x 20, $C_6:S_3$ x 19, $C_3\times A_4$ x 13, $C_3:C_{12}$ x 8, $C_6^2$ x 6

Order 32: $C_2^2\times D_4$ x 64, $C_4:D_4$ x 32, $C_2^2\wr C_2$ x 32, $C_4\times D_4$ x 16, $C_4:D_4$ x 8, $C_2^3:C_4$ x 8, $C_2^5$ x 4, $C_2^3\times C_4$ x 4, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 132, $C_3:D_4$ x 72, $C_2^2\times C_6$ x 33, $D_{12}$ x 28, $C_3\times D_4$ x 28, $S_4$ x 26, $C_4\times S_3$ x 24, $C_2\times A_4$ x 22, $C_6:C_4$ x 20, $C_2\times C_{12}$ x 14

Order 18: $C_3\times S_3$ x 18, $C_3\times C_6$ x 11, $C_3:S_3$ x 10

Order 16: $C_2\times D_4$ x 186, $C_2^4$ x 42, $C_2^2:C_4$ x 32, $C_2^2\times C_4$ x 31, $C_4:C_4$ x 8, $C_4^2$ x 4

Order 12: $D_6$ x 123, $C_2\times C_6$ x 49, $C_3:C_4$ x 12, $A_4$ x 10, $C_{12}$ x 10

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 117, $D_4$ x 96, $C_2\times C_4$ x 46

Order 6: $S_3$ x 30, $C_6$ x 27

Order 4: $C_2^2$ x 89, $C_4$ x 14

Order 3: $C_3$ x 7

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3456: $C_6:S_4^2$

Order 1728: $C_6\times \PSOPlus(4,3)$, $A_4^2:D_6$, $C_3:S_4^2$

Order 1152: $C_2\times S_4^2$, $\GL(2,\mathbb{Z}/4):D_6$

Order 864: $C_6\times A_4^2$, $C_2\times A_4:S_3^2$, $A_4^2:C_6$, $A_4^2:S_3$

Order 576: $A_4^2:C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):S_3$ x 2, $C_2^2:S_4\times C_6$, $C_2^2\times C_6:S_4$, $C_2^2:D_6^2$, $A_4^2:C_2^2$, $S_4^2$, $C_6:D_4\times A_4$, $C_6\times \GL(2,\mathbb{Z}/4)$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_2^4.S_3^2$

Order 432: $C_6^2:D_6$ x 2, $C_6:S_3\times A_4$, $C_3\times A_4^2$, $A_4:S_3^2$

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

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

Order 216: $C_3^2:S_4$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$

Order 192: $C_2^3\times S_4$ x 3, $C_2^4:D_6$ x 3, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $D_4\times S_4$ x 2, $C_2^3:S_4$, $D_6:C_2^4$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^5:C_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $C_6:S_4$ x 7, $C_6\times S_4$ x 7, $A_4\times D_6$ x 5, $C_2^2:C_6^2$ x 3, $A_4^2$ x 3, $S_3\times S_4$ x 3, $C_6^2:C_2^2$ x 2, $D_6:D_6$ x 2, $C_6:D_{12}$ x 2, $C_6^2:C_2^2$, $C_2^4:C_3^2$, $D_6^2$, $C_2^2.S_3^2$, $A_4\times C_3:C_4$, $A_4:C_{12}$

Order 128: $C_2\times D_4^2$

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

Order 96: $C_2^2\times S_4$ x 13, $C_2^3:D_6$ x 9, $D_4\times D_6$ x 9, $C_2^3\times A_4$ x 3, $\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_6$ x 3, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3\times D_6$ x 2, $C_2^3:A_4$ x 2, $C_{12}:C_2^3$ x 2, $C_2^2\times D_{12}$ x 2, $D_4\times A_4$ x 2, $C_2^3:C_{12}$ x 2, $C_{12}:D_4$ x 2, $C_2^3.D_6$ x 2, $D_6:D_4$ x 2, $C_2^2.D_{12}$ x 2, $C_2^4\times C_6$, $C_2^2:S_4$, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_4:S_4$, $C_4\times S_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_6:C_4^2$

Order 72: $C_6\times A_4$ x 9, $S_3\times D_6$ x 5, $C_3\times S_4$ x 5, $C_6:D_6$ x 4, $C_6\wr C_2$ x 4, $S_3\times A_4$ x 3, $C_3:S_4$ x 3, $C_6\times D_6$ x 2, $C_6:C_{12}$ x 2, $C_3:D_{12}$ x 2, $C_2\times C_6^2$, $C_6.D_6$

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

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

Order 48: $C_6:D_4$ x 25, $C_2\times S_4$ x 19, $C_2^2\times D_6$ x 17, $C_6\times D_4$ x 11, $S_3\times D_4$ x 10, $C_2^2\times A_4$ x 9, $C_2\times D_{12}$ x 8, $C_2^3\times C_6$ x 6, $C_4\times D_6$ x 5, $C_6.C_2^3$ x 4, $C_2^2:C_{12}$ x 4, $C_2^2:A_4$ x 2, $C_2^2\times C_{12}$ x 2, $D_6:C_4$ x 2, $C_4\times A_4$, $A_4:C_4$, $C_6.D_4$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 9, $C_6:S_3$ x 7, $C_3\times A_4$ x 7, $C_6^2$ x 3, $S_3^2$ x 3, $C_3:C_{12}$ x 2

Order 32: $C_2^2\times D_4$ x 21, $C_2^2\wr C_2$ x 7, $C_2^3:C_4$ x 5, $C_4:D_4$ x 4, $C_2^5$ x 3, $C_2^3\times C_4$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2, $C_2^2.D_4$, $C_2\times C_4^2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 41, $C_2^2\times C_6$ x 20, $C_3:D_4$ x 15, $C_2\times A_4$ x 11, $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 7, $S_4$ x 7, $D_{12}$ x 4, $C_4\times S_3$ x 3

Order 18: $C_3\times C_6$ x 7, $C_3\times S_3$ x 5, $C_3:S_3$ x 3

Order 16: $C_2\times D_4$ x 48, $C_2^4$ x 20, $C_2^2\times C_4$ x 10, $C_2^2:C_4$ x 7, $C_4:C_4$, $C_4^2$

Order 12: $D_6$ x 35, $C_2\times C_6$ x 26, $A_4$ x 6, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 45, $D_4$ x 21, $C_2\times C_4$ x 14

Order 6: $C_6$ x 15, $S_3$ x 8

Order 4: $C_2^2$ x 34, $C_4$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 1728: $C_3:S_4^2$ ($C_2$) x 4, $A_4^2:D_6$ ($C_2$) x 2, $C_6\times \PSOPlus(4,3)$ ($C_2$)

Order 864: $A_4^2:S_3$ ($C_2^2$) x 4, $A_4^2:C_6$ ($C_2^2$) x 2, $C_6\times A_4^2$ ($C_2^2$)

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

Order 432: $C_3\times A_4^2$ ($C_2^3$)

Order 288: $C_2\times C_6:S_4$ ($D_6$) x 4, $C_2^3:C_6^2$ ($D_6$) x 2, $\PSOPlus(4,3)$ ($D_6$) x 2, $C_2\times A_4^2$ ($D_6$)

Order 144: $C_2^2:C_6^2$ ($C_2\times D_6$) x 2, $C_6:S_4$ ($S_4$) x 2, $A_4^2$ ($C_2\times D_6$)

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

Order 72: $C_3:S_4$ ($C_2\times S_4$) x 4, $C_6\times A_4$ ($C_2\times S_4$) x 2

Order 48: $C_2^2\times A_4$ ($S_3\times D_6$) x 2, $C_2^3\times C_6$ ($S_3\times D_6$)

Order 36: $C_3\times A_4$ ($C_2^2\times S_4$) x 2

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: $C_2^4$ ($C_6:S_3^2$)

Order 12: $C_2\times C_6$ ($D_6\times S_4$) x 2, $A_4$ ($D_6\times S_4$) x 2

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

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

Order 4: $C_2^2$ ($C_2\times A_4:S_3^2$) x 2

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

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: $C_6\times \PSOPlus(4,3)$ ($C_2$), $A_4^2:D_6$ ($C_2$), $C_3:S_4^2$ ($C_2$)

Order 864: $C_6\times A_4^2$ ($C_2^2$), $A_4^2:C_6$ ($C_2^2$), $A_4^2:S_3$ ($C_2^2$)

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

Order 432: $C_3\times A_4^2$ ($C_2^3$)

Order 288: $C_2^3:C_6^2$ ($D_6$), $C_2\times C_6:S_4$ ($D_6$), $C_2\times A_4^2$ ($D_6$), $\PSOPlus(4,3)$ ($D_6$)

Order 144: $C_2^2:C_6^2$ ($C_2\times D_6$), $C_6:S_4$ ($S_4$), $A_4^2$ ($C_2\times D_6$)

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

Order 72: $C_6\times A_4$ ($C_2\times S_4$), $C_3:S_4$ ($C_2\times S_4$)

Order 48: $C_2^3\times C_6$ ($S_3\times D_6$), $C_2^2\times A_4$ ($S_3\times D_6$)

Order 36: $C_3\times A_4$ ($C_2^2\times S_4$)

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

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

Order 16: $C_2^4$ ($C_6:S_3^2$)

Order 12: $C_2\times C_6$ ($D_6\times S_4$), $A_4$ ($D_6\times S_4$)

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

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

Order 4: $C_2^2$ ($C_2\times A_4:S_3^2$)

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

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

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

Series

Derived series

$C_6:S_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_6:S_4^2$

$\rhd$

$C_6\times \PSOPlus(4,3)$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6:S_4^2$

$\rhd$

$C_3\times A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 22 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $78 \times 78$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $76 \times 76$ rational character table.