Abstract group 3456.oc: $D_6^2:S_4$

• Label: 3456.oc
• Order: \( 2^{7} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$D_6^2:S_4$
Order:	\(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$A_4^2:(C_6^2:D_4)$, of order \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
Composition factors:	$C_2$ x 7, $C_3$ x 3
Derived length:	$3$

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	447	296	576	1416	720	3456
Conjugacy classes	1	23	7	12	32	15	90
Divisions	1	23	7	12	32	15	90
Autjugacy classes	1	11	5	3	13	3	36

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	8	12	24	6	28	1	10	1	90
Irr. rational chars.	8	12	24	6	28	1	10	1	90

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	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

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

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 108168 subgroups in 6076 conjugacy classes, 78 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $D_6^2:S_4$
Commutator:	$G' \simeq$ $C_6^2:A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_6^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2\times D_6$
Radical:	$R \simeq$ $D_6^2:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times D_6$
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: $D_6^2:S_4$

Order 1728: $(C_6\times D_6):S_4$ x 2, $C_3\times C_2^4:S_3^2$ x 2, $D_6^2:A_4$, $C_6^2:(C_2\times S_4)$, $C_6^2:(C_2\times S_4)$

Order 1152: $C_2^5:S_3^2$ x 2, $D_6^2:D_4$

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

Order 576: $C_2^4:S_3^2$ x 11, $D_6^2:C_2^2$ x 6, $D_6^2:C_2^2$ x 6, $C_4:D_6^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_4$ x 3, $C_2^2:A_4\times D_6$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_2^2:S_4\times C_6$ x 2, $C_6^2.C_2^4$ x 2, $C_2^4:S_3^2$ x 2, $C_2^2\times D_6^2$, $C_6^2.C_2^4$, $C_2^4:S_3^2$

Order 432: $A_4:S_3^2$ x 6, $C_6^2:D_6$ x 6, $A_4\times S_3^2$ x 4, $A_4:S_3^2$ x 3, $C_6^2:D_6$ x 3, $C_6^2:A_4$

Order 384: $C_2^5:D_6$ x 2, $C_2^2\wr S_3$

Order 288: $D_4\times S_3^2$ x 24, $C_2\times D_6^2$ x 17, $D_6^2:C_2$ x 12, $D_6:D_{12}$ x 12, $D_6.D_{12}$ x 12, $(C_2\times C_6):S_4$ x 8, $S_3\times C_2^2:A_4$ x 7, $C_3\times C_2^2:S_4$ x 7, $C_6^2:C_2^3$ x 6, $D_6^2:C_2$ x 6, $D_6\times D_{12}$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_6^2.C_2^3$ x 6, $D_6^2:C_2$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_6^2:D_4$ x 6, $C_2^3.S_3^2$ x 6, $D_6:D_{12}$ x 6, $D_6^2.C_2$ x 6, $D_6\times S_4$ x 6, $D_{12}:D_6$ x 3, $C_2.D_6^2$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $C_2^3.S_3^2$ x 3, $C_6^2:C_2^3$ x 3, $C_6^2:D_4$ x 2, $C_6^2:C_2^3$ x 2, $C_2^5:C_3^2$ x 2, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:C_2^3$

Order 216: $C_6^2:C_6$ x 8, $C_3^2:S_4$ x 6, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 3, $C_3^2\times S_4$ x 3, $S_3^3$

Order 192: $C_2^4:D_6$ x 33, $C_2^3:S_4$ x 8, $C_{12}:C_2^4$ x 6, $C_2^4:D_6$ x 6, $C_2^3:D_{12}$ x 6, $C_2^4.D_6$ x 6, $D_6\times C_2^4$ x 2, $C_2^5:C_6$ x 2, $C_2^4:D_6$ x 2, $C_2^2\wr C_3$

Order 144: $D_6^2$ x 130, $D_6:D_6$ x 48, $S_3\times S_4$ x 33, $C_{12}:D_6$ x 24, $D_6:D_6$ x 24, $S_3\times D_{12}$ x 24, $C_6^2:C_2^2$ x 18, $D_6:C_{12}$ x 12, $C_6.D_{12}$ x 12, $C_{12}:D_6$ x 12, $C_6^2:C_2^2$ x 12, $C_6:D_{12}$ x 12, $C_{12}:D_6$ x 12, $C_4\times S_3^2$ x 12, $C_6^2:C_2^2$ x 9, $A_4\times D_6$ x 8, $C_6.D_{12}$ x 6, $C_6^2:C_4$ x 6, $C_6.D_{12}$ x 6, $C_2^4:C_3^2$ x 6, $C_6:S_4$ x 6, $C_6\times S_4$ x 6, $C_6^2:C_2^2$ x 6, $C_6\times D_{12}$ x 6, $C_{12}\times D_6$ x 6, $D_6:D_6$ x 6, $D_6.D_6$ x 6, $C_4:C_6^2$ x 3, $C_6:D_{12}$ x 3, $C_{12}:D_6$ x 3, $C_2^2.S_3^2$ x 3, $C_6^2:C_4$ x 3, $C_6^2:C_4$ x 3, $C_2^2\times C_6^2$

Order 128: $C_2^4:D_4$

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

Order 96: $D_4\times D_6$ x 99, $C_2^2:D_{12}$ x 51, $D_6.D_4$ x 51, $D_6:D_4$ x 51, $C_2^3\times D_6$ x 39, $C_2^4:C_6$ x 17, $C_2^4:S_3$ x 17, $C_2^3:D_6$ x 12, $C_2^2.D_{12}$ x 12, $C_2^2:S_4$ x 9, $C_{12}:C_2^3$ x 6, $C_2^2\times D_{12}$ x 6, $C_{12}:C_2^3$ x 6, $C_2^3:C_{12}$ x 6, $C_2^3.D_6$ x 6, $C_2^3:A_4$ x 5, $C_2^2\times S_4$ x 3, $C_2^4\times C_6$ x 2

Order 72: $S_3\times D_6$ x 220, $C_6\times D_6$ x 80, $C_6:D_6$ x 40, $S_3\times A_4$ x 28, $C_3:S_4$ x 24, $C_6\wr C_2$ x 24, $C_3:D_{12}$ x 24, $C_3\times S_4$ x 21, $C_6^2:C_2$ x 12, $C_3\times D_{12}$ x 12, $S_3\times C_{12}$ x 12, $D_6:S_3$ x 12, $C_6.D_6$ x 12, $C_6\times A_4$ x 8, $D_4\times C_3^2$ x 6, $C_3:D_{12}$ x 6, $C_{12}:S_3$ x 6, $C_6:C_{12}$ x 6, $C_6.D_6$ x 6, $C_2\times C_6^2$ x 5, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3

Order 64: $C_2^3:D_4$ x 24, $D_4\times C_2^3$ x 3, $C_2^4:C_4$ x 3, $C_2^6$

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$

Order 48: $C_2^2\times D_6$ x 305, $S_3\times D_4$ x 204, $C_6:D_4$ x 102, $D_6:C_4$ x 54, $C_6\times D_4$ x 51, $C_2\times D_{12}$ x 51, $C_4\times D_6$ x 51, $C_2^2:C_{12}$ x 27, $C_6.D_4$ x 27, $C_2\times S_4$ x 24, $C_2^3\times C_6$ x 21, $C_2^2\times C_{12}$ x 6, $C_6.C_2^3$ x 6, $C_2^2:A_4$ x 4, $C_2^2\times A_4$ x 4

Order 36: $S_3^2$ x 74, $C_6\times S_3$ x 70, $C_6:S_3$ x 35, $C_3\times A_4$ x 24, $C_6^2$ x 14, $C_3:C_{12}$ x 6, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 3

Order 32: $C_2^2\wr C_2$ x 64, $C_2^2\times D_4$ x 60, $C_2^3:C_4$ x 36, $C_2^5$ x 20, $C_2^3\times C_4$ x 3

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 637, $C_3:D_4$ x 108, $C_2^2\times C_6$ x 95, $D_{12}$ x 54, $C_4\times S_3$ x 54, $C_3\times D_4$ x 54, $C_2\times C_{12}$ x 27, $C_6:C_4$ x 27, $S_4$ x 27, $C_2\times A_4$ x 20

Order 18: $C_3\times S_3$ x 26, $C_3:S_3$ x 14, $C_3\times C_6$ x 7

Order 16: $C_2\times D_4$ x 192, $C_2^4$ x 158, $C_2^2:C_4$ x 48, $C_2^2\times C_4$ x 30

Order 12: $D_6$ x 299, $C_2\times C_6$ x 109, $A_4$ x 16, $C_{12}$ x 15, $C_3:C_4$ x 15

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 343, $D_4$ x 96, $C_2\times C_4$ x 48

Order 6: $S_3$ x 39, $C_6$ x 32

Order 4: $C_2^2$ x 176, $C_4$ x 12

Order 3: $C_3$ x 7

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3456: $D_6^2:S_4$

Order 1728: $D_6^2:A_4$, $C_6^2:(C_2\times S_4)$, $(C_6\times D_6):S_4$, $C_6^2:(C_2\times S_4)$, $C_3\times C_2^4:S_3^2$

Order 1152: $C_2^5:S_3^2$, $D_6^2:D_4$

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

Order 576: $C_2^4:S_3^2$ x 6, $C_2^2\times D_6^2$, $C_2^2:A_4\times D_6$, $(C_2^2\times C_6):S_4$, $C_2^2:S_4\times C_6$, $C_4:D_6^2$, $C_6^2.C_2^4$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_2^4:S_3^2$, $D_6^2:C_2^2$, $D_6^2:C_2^2$, $D_6^2:C_2^2$, $D_6^2:C_2^2$, $D_6^2:C_4$

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

Order 384: $C_2^2\wr S_3$, $C_2^5:D_6$

Order 288: $C_2\times D_6^2$ x 7, $D_4\times S_3^2$ x 6, $(C_2\times C_6):S_4$ x 5, $S_3\times C_2^2:A_4$ x 4, $C_3\times C_2^2:S_4$ x 4, $D_6^2:C_2$ x 2, $D_6^2:C_2$ x 2, $C_6^2:C_2^3$, $D_{12}:D_6$, $D_6\times D_{12}$, $C_2.D_6^2$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $D_6^2:C_2$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $D_6:D_{12}$, $D_6:D_{12}$, $D_6^2.C_2$, $D_6.D_{12}$, $C_6^2:C_2^3$, $C_6^2:C_2^3$, $C_2^5:C_3^2$, $D_6\times S_4$, $C_6^2:C_2^3$

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

Order 192: $C_2^4:D_6$ x 9, $C_2^3:S_4$ x 5, $D_6\times C_2^4$, $C_2^2\wr C_3$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $D_6^2$ x 31, $D_6:D_6$ x 8, $D_6:D_6$ x 7, $S_3\times S_4$ x 6, $C_6^2:C_2^2$ x 5, $C_6^2:C_2^2$ x 5, $C_2^4:C_3^2$ x 4, $C_{12}:D_6$ x 4, $C_{12}:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_{12}:D_6$ x 3, $C_4\times S_3^2$ x 3, $A_4\times D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $C_6:D_{12}$ x 2, $C_6.D_{12}$, $C_6^2:C_4$, $D_6:C_{12}$, $C_6.D_{12}$, $C_6.D_{12}$, $C_2^2\times C_6^2$, $C_6:S_4$, $C_6\times S_4$, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$, $D_6:D_6$, $C_2^2.S_3^2$, $D_6.D_6$, $C_6^2:C_4$, $C_6^2:C_4$

Order 128: $C_2^4:D_4$

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

Order 96: $D_4\times D_6$ x 17, $C_2^3\times D_6$ x 12, $C_2^2:D_{12}$ x 7, $D_6.D_4$ x 7, $D_6:D_4$ x 7, $C_2^2:S_4$ x 6, $C_2^4:C_6$ x 5, $C_2^4:S_3$ x 5, $C_2^3:A_4$ x 3, $C_2^3:D_6$ x 2, $C_2^4\times C_6$, $C_2^2\times S_4$, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^3.D_6$, $C_2^2.D_{12}$

Order 72: $S_3\times D_6$ x 40, $C_6:D_6$ x 15, $C_6\times D_6$ x 15, $S_3\times A_4$ x 8, $C_3:S_4$ x 5, $C_3\times S_4$ x 4, $C_6^2:C_2$ x 4, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $C_2\times C_6^2$ x 3, $C_6\times A_4$ x 2, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $C_6.D_6$ x 2, $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$

Order 64: $C_2^3:D_4$ x 8, $C_2^6$, $D_4\times C_2^3$, $C_2^4:C_4$

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

Order 48: $C_2^2\times D_6$ x 57, $S_3\times D_4$ x 36, $C_6:D_4$ x 18, $C_6\times D_4$ x 9, $C_2\times D_{12}$ x 9, $C_4\times D_6$ x 9, $C_2^3\times C_6$ x 7, $C_2\times S_4$ x 5, $D_6:C_4$ x 5, $C_2^2:C_{12}$ x 4, $C_6.D_4$ x 4, $C_2^2:A_4$ x 3, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 36: $S_3^2$ x 22, $C_6\times S_3$ x 14, $C_6:S_3$ x 13, $C_3\times A_4$ x 8, $C_6^2$ x 6, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$ x 14, $C_2^2\wr C_2$ x 13, $C_2^5$ x 9, $C_2^3:C_4$ x 6, $C_2^3\times C_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 105, $C_2^2\times C_6$ x 21, $C_3:D_4$ x 20, $D_{12}$ x 10, $C_4\times S_3$ x 10, $C_3\times D_4$ x 10, $C_2\times A_4$ x 6, $S_4$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 5

Order 18: $C_3\times S_3$ x 12, $C_3:S_3$ x 9, $C_3\times C_6$ x 4

Order 16: $C_2^4$ x 42, $C_2\times D_4$ x 40, $C_2^2:C_4$ x 7, $C_2^2\times C_4$ x 7

Order 12: $D_6$ x 63, $C_2\times C_6$ x 25, $A_4$ 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 72, $D_4$ x 20, $C_2\times C_4$ x 11

Order 6: $S_3$ x 18, $C_6$ x 13

Order 4: $C_2^2$ x 46, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3456: $D_6^2:S_4$ ($C_1$)

Order 1728: $(C_6\times D_6):S_4$ ($C_2$) x 2, $C_3\times C_2^4:S_3^2$ ($C_2$) x 2, $D_6^2:A_4$ ($C_2$), $C_6^2:(C_2\times S_4)$ ($C_2$), $C_6^2:(C_2\times S_4)$ ($C_2$)

Order 864: $(C_6\times D_6):A_4$ ($C_2^2$) x 2, $C_6^2:S_4$ ($C_2^2$) x 2, $C_6^2:(C_2\times A_4)$ ($C_2^2$), $C_6^2:S_4$ ($C_2^2$), $C_6^2:S_4$ ($C_2^2$)

Order 576: $C_2^4:S_3^2$ ($S_3$) x 2, $C_2^2\times D_6^2$ ($S_3$)

Order 432: $C_6^2:A_4$ ($C_2^3$)

Order 288: $C_6^2:C_2^3$ ($D_6$) x 2, $S_3\times C_2^2:A_4$ ($D_6$) x 2, $(C_2\times C_6):S_4$ ($D_6$) x 2, $C_3\times C_2^2:S_4$ ($D_6$) x 2, $C_6^2:C_2^3$ ($D_6$)

Order 144: $D_6^2$ ($S_4$) x 3, $C_2^4:C_3^2$ ($C_2\times D_6$) x 2, $C_2^2\times C_6^2$ ($C_2\times D_6$)

Order 96: $C_2^3\times D_6$ ($S_3^2$) x 2, $C_2^2:S_4$ ($S_3^2$)

Order 72: $C_6\times D_6$ ($C_2\times S_4$) x 6, $C_6:D_6$ ($C_2\times S_4$) x 3

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

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

Order 24: $C_2\times D_6$ ($S_3\times S_4$) x 6

Order 18: $C_3\times S_3$ ($C_2^3:S_4$) x 2, $C_3:S_3$ ($C_2^3:S_4$)

Order 16: $C_2^4$ ($S_3^3$)

Order 12: $C_2\times C_6$ ($D_6\times S_4$) x 6

Order 9: $C_3^2$ ($C_2^2\wr S_3$)

Order 6: $S_3$ ($C_2^4:S_3^2$) x 2

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

Order 3: $C_3$ ($C_2^5:S_3^2$) x 2

Order 1: $C_1$ ($D_6^2:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3456: $D_6^2:S_4$ ($C_1$)

Order 1728: $D_6^2:A_4$ ($C_2$), $C_6^2:(C_2\times S_4)$ ($C_2$), $(C_6\times D_6):S_4$ ($C_2$), $C_6^2:(C_2\times S_4)$ ($C_2$), $C_3\times C_2^4:S_3^2$ ($C_2$)

Order 864: $C_6^2:(C_2\times A_4)$ ($C_2^2$), $(C_6\times D_6):A_4$ ($C_2^2$), $C_6^2:S_4$ ($C_2^2$), $C_6^2:S_4$ ($C_2^2$), $C_6^2:S_4$ ($C_2^2$)

Order 576: $C_2^2\times D_6^2$ ($S_3$), $C_2^4:S_3^2$ ($S_3$)

Order 432: $C_6^2:A_4$ ($C_2^3$)

Order 288: $C_6^2:C_2^3$ ($D_6$), $C_6^2:C_2^3$ ($D_6$), $S_3\times C_2^2:A_4$ ($D_6$), $(C_2\times C_6):S_4$ ($D_6$), $C_3\times C_2^2:S_4$ ($D_6$)

Order 144: $C_2^2\times C_6^2$ ($C_2\times D_6$), $C_2^4:C_3^2$ ($C_2\times D_6$), $D_6^2$ ($S_4$)

Order 96: $C_2^3\times D_6$ ($S_3^2$), $C_2^2:S_4$ ($S_3^2$)

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

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

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

Order 24: $C_2\times D_6$ ($S_3\times S_4$)

Order 18: $C_3:S_3$ ($C_2^3:S_4$), $C_3\times S_3$ ($C_2^3:S_4$)

Order 16: $C_2^4$ ($S_3^3$)

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

Order 9: $C_3^2$ ($C_2^2\wr S_3$)

Order 6: $S_3$ ($C_2^4:S_3^2$)

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

Order 3: $C_3$ ($C_2^5:S_3^2$)

Order 1: $C_1$ ($D_6^2:S_4$)

Series

Derived series

$D_6^2:S_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$D_6^2:S_4$

$\rhd$

$C_6^2:(C_2\times S_4)$

$\rhd$

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

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^4:C_3^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$D_6^2:S_4$

$\rhd$

$C_6^2:A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 6 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 $90 \times 90$ rational character table. Alternatively, you may search for characters of this group with desired properties.