Abstract group 41472.dq: $C_6^2:S_4\wr C_2$

• Label: 41472.dq
• Order: \( 2^{9} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^2:S_4\wr C_2$
Order:	\(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_{1191}:C_{44}$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \)
Composition factors:	$C_2$ x 9, $C_3$ x 4
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	8	12
Elements	1	2079	728	10656	12744	5184	10080	41472
Conjugacy classes	1	21	14	20	105	2	26	189
Divisions	1	21	14	20	105	2	26	189
Autjugacy classes	1	13	6	10	38	1	7	76

Dimension	1	2	4	6	8	9	12	18	24	36	72
Irr. complex chars.	8	6	48	16	28	8	40	6	16	12	1	189
Irr. rational chars.	8	6	48	16	28	8	40	6	16	12	1	189

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{4}=c^{6}=d^{6}=e^{6}=f^{2}=g^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,2,3,4)(5,6,8,7)(9,11,13)(10,12)(15,16)(17,18), (6,7)(9,10)(11,12)(13,14)(16,17), (1,2,3,5)(4,6,8,7)(11,13)(15,16,18,17)\rangle$
Transitive group:	36T15348	36T15350			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $(S_4\wr C_2)$	$(C_3^2:S_4^2)$ $\,\rtimes\,$ $D_4$	$C_3^2$ $\,\rtimes\,$ $(S_4^2:D_4)$	$A_4^2$ $\,\rtimes\,$ $(D_6\wr C_2)$	all 18
Trans. wreath product:	$(C_2 imes C_3:S_4)$ $\ \wr\ $$C_2$
Non-split product:	$C_2^5$ . $(C_2\times C_3^4:D_4)$	$(C_3\times C_6)$ . $(S_4^2:C_2^2)$	$C_2$ . $(C_3^2:S_4^2:C_2^2)$	$((C_3\times A_4^2).D_6)$ . $C_2^3$	all 9

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 3232098 subgroups in 32162 conjugacy classes, 47 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^2:S_4^2:C_2^2$
Commutator:	$G' \simeq$ $(C_3\times A_4^2).D_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3^2:S_4^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_6^2$	$G/\operatorname{Fit} \simeq$ $\SOPlus(4,2)$
Radical:	$R \simeq$ $C_6^2:S_4\wr C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{soc} \simeq$ $S_3^2:C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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

All subgroups of index up to 96 (order at least 432) are shown, as well as all normal subgroups of any index.

Order 41472: $C_6^2:S_4\wr C_2$

Order 20736: $C_3^2:S_4^2:C_2^2$ x 2, $C_3^2:S_4^2:C_4$ x 2, $C_2\times A_4^2.C_3^2.C_2^3$, $C_6^2:\POPlus(4,3)$, $A_4^2:(C_6^2:C_4)$

Order 10368: $C_2\times (C_3\times A_4^2).D_6$ x 5, $A_4^2:\SOPlus(4,2)$ x 4, $A_4^2.C_3^2.C_2^3$ x 2, $C_2^5:(C_3^4:C_4)$ x 2, $C_2^2\times (C_3\times A_4^2).C_6$, $C_2\times (C_3\times A_4^2).D_6$, $(C_3\times A_4^2):D_{12}$, $(C_6\times A_4^2).D_6$, $C_2\times A_4^2:S_3^2$, $(C_6\times A_4)\wr C_2$

Order 6912: $C_2^2\times C_3:S_4^2$ x 3, $A_4^2:(S_3\times D_4)$ x 3, $C_2\times (C_3\times C_6\times A_4).C_2^4$, $(C_2^4\times C_6^2):D_6$

Order 5184: $C_3^2:S_4^2$ x 10, $C_2\times (C_3\times A_4^2).C_6$ x 6, $C_2\times A_4^2.C_3:S_3$ x 3, $(C_3^2\times A_4^2):C_4$ x 2, $A_4^2:S_3^2$ x 2, $C_2^2\times A_4^2.C_3^2$, $C_2\times (C_3^3\times A_4).C_2^3$, $(C_3\times A_4^2).D_6$, $C_6\times A_4^2:S_3$, $(C_3\times A_4^2):C_{12}$

Order 4608: $S_4^2:D_4$ x 3, $C_6^2.C_2^4.C_2^3$

Order 3456: $C_2\times A_4^2.D_6$ x 36, $(C_3\times C_6\times A_4).C_2^4$ x 22, $C_2\times A_4^2:D_6$ x 12, $C_2^2:C_6^2\times S_4$ x 3, $C_2\times A_4^2:D_6$ x 3, $A_4^2:D_{12}$ x 3, $(C_3\times A_4^2):D_4$ x 3, $A_4^2:(C_4\times S_3)$ x 3, $C_2\times A_4^2:D_6$ x 3, $(C_3\times A_4^2):D_4$ x 3, $C_3\times A_4^2:D_4$ x 3, $C_2\times A_4.C_6^2:C_2.C_2$ x 2, $C_2\times A_4.D_6^2$, $C_2\times (C_6^2\times A_4).C_2^2$, $C_2^2.C_6^3.C_2^2$, $C_2\times (C_3\times C_6\times A_4).C_2^3$, $C_2\times C_2^4.C_3^3:C_2.C_2$, $C_2^2\times (C_6^2\times A_4).C_2$, $C_2\times (C_3\times C_6\times A_4).C_2^3$, $C_2\times A_4.(C_2\times D_6:S_3)$, $C_6^2.(C_4\times S_4)$, $(C_2^3\times C_6^2):D_6$, $(C_2^2\times C_6^2):D_{12}$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_6^2):C_6$

Order 2592: $C_6^3:D_6$ x 12, $A_4\times C_3^2:S_4$ x 4, $(A_4\times C_6^2):S_3$ x 3, $C_2\times A_4^2.C_3^2$ x 2, $A_4.(C_6^2\times S_3)$, $C_2^2\times (C_3^3\times A_4).C_2$, $C_2\times (C_3^2\times A_4).D_6$, $(C_3\times A_4)\wr C_2$, $C_6^2:\SOPlus(4,2)$

Order 2304: $C_2^2\times S_4^2$ x 6, $C_2\wr S_3^2$ x 6, $S_4^2:C_4$ x 6, $(C_2^2\times A_4^2):C_4$ x 6, $S_4^2:C_2^2$ x 6, $C_6^2.C_2^4.C_2^2$ x 4, $C_6^2.C_2^4.C_2^2$ x 4, $C_2^4:D_6^2$ x 3, $C_2^6:S_3^2$ x 3, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^5.C_2$, $C_6^2.C_2^6$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^5.C_2$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$, $C_{12}:(C_2^3\times S_4)$, $C_2\wr C_6:S_3$

Order 1728: $A_4^2:D_6$ x 66, $C_6^2:(C_2\times S_4)$ x 64, $C_3:S_4^2$ x 48, $A_4:D_6^2$ x 23, $A_4^2:D_6$ x 18, $C_6\times A_4\times S_4$ x 18, $C_2\times C_6^2:S_4$ x 14, $C_2^6:C_3^3$ x 12, $A_4\times C_6^2:C_2^2$ x 12, $C_2\times C_6^2:S_4$ x 12, $C_2\times C_3^2:\GL(2,\mathbb{Z}/4)$ x 12, $C_2\times C_6^2:D_{12}$ x 12, $C_6^2:(C_4\times D_6)$ x 12, $A_4^2:D_6$ x 6, $C_2\times C_6^2:S_4$ x 4, $C_6:D_6\times S_4$ x 3, $A_4^2:D_6$ x 3, $A_4\wr C_2\times C_6$ x 3, $(C_3\times A_4^2):C_4$ x 3, $A_4^2:C_{12}$ x 3, $(C_6\times D_6):S_4$ x 2, $C_2^3:C_6^3$, $C_2^6:C_3^3$, $S_3\times C_2^3:C_6^2$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_6^3:C_2^3$, $A_4\times C_6^2:C_4$, $(C_2\times C_6^2).S_4$, $C_2\times C_6^2.S_4$, $C_6^2:(C_2\times S_4)$, $(C_2^3\times C_6^2):C_6$, $(C_2^2\times C_6^2):C_{12}$

Order 1296: $C_6^2:S_3^2$ x 16, $C_6^3:C_6$ x 6, $C_6^3:S_3$ x 6, $C_6^3:S_3$ x 6, $C_2\times C_3^4:D_4$ x 2, $(C_3^3\times C_6).D_4$ x 2, $C_6^3:C_6$, $C_6^2:S_3^2$, $C_3^2\times A_4^2$, $C_6^2:S_3^2$, $(C_3^2\times C_6^2):C_4$

Order 1152: $C_2^5:S_3^2$ x 72, $C_2\times S_4^2$ x 57, $A_4^2:C_2^3$ x 27, $D_6^2.C_2^3$ x 26, $C_6:S_4\times D_4$ x 24, $D_6^2:D_4$ x 16, $C_2\times A_4^2:C_4$ x 12, $S_4\wr C_2$ x 12, $A_4^2:C_2^3$ x 9, $C_2\times C_6:\GL(2,\mathbb{Z}/4)$ x 9, $A_4^2:D_4$ x 9, $A_4^2:D_4$ x 9, $(C_2^3\times C_6):S_4$ x 8, $A_4^2:C_2^3$ x 6, $C_2^5.S_3^2$ x 6, $C_2\times C_6^2.C_2^2:C_4$ x 4, $C_2\times C_3^2.C_2\wr C_4$ x 4, $C_2\times C_3^2.C_2\wr C_4$ x 4, $C_6\times C_2^3\times S_4$ x 3, $C_2^3:D_6^2$ x 3, $C_2^3:D_6\times A_4$ x 3, $C_3\times C_2\wr S_3$ x 3, $C_2^3.D_6^2$ x 3, $C_2^3.D_6^2$ x 3, $C_2^3.D_6^2$ x 3, $C_2^5:S_3^2$ x 3, $C_3:C_2\wr C_6$ x 3, $C_2^5.S_3^2$ x 3, $C_2^5.S_3^2$ x 3, $C_2^5.S_3^2$ x 3, $D_6^2.C_2^3$ x 2, $C_2\times C_6^2.(C_2\times D_4)$ x 2, $C_2\times C_3^2.D_4:D_4$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_3^2:C_4.C_2^4.C_2$ x 2, $C_3^2.(C_2^5.C_2).C_2$ x 2, $C_6^2.(C_4\times D_4)$ x 2, $C_3^2:C_4.(C_4\times D_4)$ x 2, $C_2^3\times C_6:S_4$ x 2, $(C_2\times D_6:S_3).C_2^3$, $D_6^2:D_4$, $(C_2\times C_6^2).C_2^4$, $(C_2^3\times C_3:S_3).C_2^3$, $C_6^2.(C_4\times D_4)$, $C_3^2:C_4.(C_4\times D_4)$, $C_6^2.(C_2^2\times C_4).C_2$, $(C_6\times C_6.D_4).C_2^2$, $C_6^2.(C_2^2\times C_4).C_2$, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_2^2\times C_{12}:S_4$, $C_2^2\times C_{12}:S_4$, $(C_2^3\times C_6):S_4$, $(C_2^3\times C_6):S_4$, $(C_2^2\times C_{12}):S_4$, $(C_2^2\times C_{12}):S_4$, $C_2^5:C_6^2$, $C_2^5.C_6^2$, $C_6^2.C_2^5$, $C_6^2.C_2^5$, $C_6^2.C_2^5$, $C_2^2\times C_3^2:(C_4:D_4)$, $C_2^2\times C_2^2:(C_3^2:D_4)$, $C_2^2\times (C_6\times S_3):D_4$, $C_2^2\times (C_2^3\times S_3):S_3$, $C_2^2\times D_6^2:C_2$, $C_2^2\times (C_2^3\times C_6):S_3$, $C_2^2\times C_3:C_4\times C_3:D_4$

Order 864: $S_3\times C_6:S_4$ x 101, $C_6^2:S_4$ x 46, $A_4^2:S_3$ x 45, $C_6:S_3\times S_4$ x 36, $C_6^2:S_4$ x 33, $C_6\times A_4^2$ x 33, $C_6^3:C_2^2$ x 24, $C_6\wr C_2\times A_4$ x 17, $C_6^2:D_{12}$ x 17, $C_6^2:S_4$ x 16, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 16, $C_3^2:(C_4\times S_4)$ x 16, $C_6\times A_4\times D_6$ x 13, $A_4^2:S_3$ x 12, $A_4^2:C_6$ x 12, $C_2^2:C_6^3$ x 7, $C_6:C_{12}\times A_4$ x 6, $C_6^2.S_4$ x 6, $C_6:D_6\times A_4$ x 3, $C_6^2:S_4$ x 3, $S_4\times C_6^2$ x 3, $A_4^2:S_3$ x 3, $A_4^2:C_6$ x 3, $C_2^5:C_3^3$ x 2, $C_6^3:C_2^2$, $C_6:D_6^2$, $C_6^2:S_4$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:(C_4\times S_4)$, $(C_6\times D_6):A_4$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$ x 3, $C_2\wr D_6$ x 3, $C_2^6:D_6$, $C_2^6:D_6$

Order 648: $C_3^4:C_2^3$ x 8, $C_3\times C_6^2:C_6$ x 4, $C_3^3:S_4$ x 4, $C_3^3:S_4$ x 4, $C_3^4:D_4$ x 4, $C_6^3:C_3$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$

Order 576: $C_2^4:S_3^2$ x 192, $A_4^2:C_2^2$ x 144, $C_2^4:S_3^2$ x 136, $C_6:\GL(2,\mathbb{Z}/4)$ x 108, $S_4^2$ x 78, $C_2^2:D_6^2$ x 69, $D_4\times C_3:S_4$ x 65, $A_4^2:C_2^2$ x 54, $C_2^2:D_6^2$ x 43, $C_2^2\times C_6:S_4$ x 42, $C_2^2\times C_6\times S_4$ x 42, $(C_2^2\times C_6):S_4$ x 41, $C_6:D_4\times A_4$ x 36, $C_6:\GL(2,\mathbb{Z}/4)$ x 36, $C_2^4.S_3^2$ x 36, $C_2^4.S_3^2$ x 36, $C_6^2:C_2^4$ x 28, $C_2^2\times A_4^2$ x 27, $C_2^2.D_6^2$ x 24, $C_2^2.D_6^2$ x 24, $C_2^2.D_6^2$ x 24, $C_2^2.D_6^2$ x 24, $C_6^2.C_2^4$ x 14, $C_2^4:S_3^2$ x 14, $C_2^2.D_6^2$ x 14, $\POPlus(4,3)$ x 12, $A_4^2:C_4$ x 12, $C_2^4:C_6^2$ x 12, $C_2\times C_{12}:S_4$ x 12, $C_2\times C_{12}:S_4$ x 12, $C_2^4:C_6^2$ x 9, $A_4^2:C_2^2$ x 9, $A_4^2:C_4$ x 9, $C_2^4:C_6^2$ x 8, $C_2^2\times C_6.S_4$ x 8, $C_2^3.\SOPlus(4,2)$ x 8, $D_6^2:C_2^2$ x 8, $C_2^4:S_3^2$ x 8, $(C_2^2\times C_6^2):C_4$ x 8, $C_2^3.\SOPlus(4,2)$ x 8, $D_6^2:C_4$ x 8, $C_2^4:S_3^2$ x 6, $D_6^2:C_2^2$ x 4, $D_6^2:C_4$ x 4, $C_2^2\times A_4\times D_6$ x 3, $C_2^2:S_4\times C_6$ x 3, $C_6.C_2^3\times A_4$ x 3, $C_3\times C_2^3.S_4$ x 3, $\GL(2,\mathbb{Z}/4):S_3$ x 3, $C_6^2.C_2^4$ x 3, $(C_2^2\times C_6^2).C_4$ x 3, $(C_2^2\times C_6^2):C_4$ x 3, $C_2^4:(C_6\times S_3)$ x 3, $(C_2^3\times C_6):C_{12}$ x 3, $C_2^4.S_3^2$ x 3, $(C_2^3\times C_6):C_{12}$ x 3, $C_6^2:C_2^4$ x 2, $C_2^2.D_6^2$ x 2, $C_2\times C_6^2.D_4$ x 2, $C_6^2.C_2^4$ x 2, $C_3^2:D_4^2$ x 2, $C_2^2.D_6^2$ x 2, $C_2^2.D_6^2$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_6^2:\SD_{16}$ x 2, $C_6^2:D_8$ x 2, $C_2\times C_6^2.D_4$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $D_6^2:C_4$ x 2, $C_2^4.S_3^2$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $C_6^2.C_4^2$ x 2, $C_2^4\times C_6^2$, $C_6^2:C_2^4$, $C_2^2\times D_6^2$, $C_6^2:C_2^4$, $C_6^2.C_2^4$, $C_2^4.C_6^2$, $C_2^4:S_3^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $(C_2^2\times C_6).S_4$, $C_2^4:C_6^2$, $C_2^4:S_3^2$, $C_2^4:S_3^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_6^2:C_4^2$, $(C_2^2\times C_6^2):C_4$, $C_6^2:\OD_{16}$, $C_3\times C_2^4:C_{12}$

Order 512: $D_4^2:D_4$

Order 432: $C_6^2:D_6$ x 152, $A_4:S_3^2$ x 82, $C_6^2:D_6$ x 64, $C_6^2:D_6$ x 57, $C_6^2:D_6$ x 48, $A_4\times C_6^2$ x 35, $C_6\times S_3\times A_4$ x 35, $C_3:D_6^2$ x 23, $C_6:S_3\times A_4$ x 18, $A_4:C_6^2$ x 18, $C_6^3:C_2$ x 14, $C_3\times A_4^2$ x 12, $C_6^3:C_2$ x 12, $D_6:C_6^2$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2:C_{12}$ x 5, $C_6^2:D_6$ x 4, $(C_3^2\times A_4):C_4$ x 4, $C_2\times C_6^3$, $C_6^3:C_2$, $D_6\times C_6^2$, $C_6^2:A_4$, $C_6^3.C_2$, $C_6^2:C_{12}$, $C_3\times C_6.S_4$

Order 288: $D_6\wr C_2$ x 11, $C_2\times A_4^2$ x 3, $C_2^3\times C_6^2$

Order 144: $A_4^2$ x 3, $C_2^2\times C_6^2$

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 20

Order 64: $C_2^6$

Order 36: $C_6^2$

Order 32: $C_2^2\wr C_2$ x 8, $C_2^5$

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $D_4$ x 15

Order 4: $C_2^2$ x 57

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 96 (order at least 432) are shown, as well as all normal subgroups of any index.

Order 41472: $C_6^2:S_4\wr C_2$

Order 20736: $C_2\times A_4^2.C_3^2.C_2^3$, $C_3^2:S_4^2:C_2^2$, $C_3^2:S_4^2:C_4$, $C_6^2:\POPlus(4,3)$, $A_4^2:(C_6^2:C_4)$

Order 10368: $C_2\times (C_3\times A_4^2).D_6$ x 2, $A_4^2.C_3^2.C_2^3$, $C_2^2\times (C_3\times A_4^2).C_6$, $C_2\times (C_3\times A_4^2).D_6$, $(C_3\times A_4^2):D_{12}$, $(C_6\times A_4^2).D_6$, $C_2\times A_4^2:S_3^2$, $C_2^5:(C_3^4:C_4)$, $(C_6\times A_4)\wr C_2$, $A_4^2:\SOPlus(4,2)$

Order 6912: $C_2\times (C_3\times C_6\times A_4).C_2^4$, $C_2^2\times C_3:S_4^2$, $A_4^2:(S_3\times D_4)$, $(C_2^4\times C_6^2):D_6$

Order 5184: $C_3^2:S_4^2$ x 3, $C_2\times A_4^2.C_3:S_3$ x 2, $C_2\times (C_3\times A_4^2).C_6$ x 2, $C_2^2\times A_4^2.C_3^2$, $C_2\times (C_3^3\times A_4).C_2^3$, $(C_3\times A_4^2).D_6$, $C_6\times A_4^2:S_3$, $(C_3\times A_4^2):C_{12}$, $(C_3^2\times A_4^2):C_4$, $A_4^2:S_3^2$

Order 4608: $C_6^2.C_2^4.C_2^3$, $S_4^2:D_4$

Order 3456: $(C_3\times C_6\times A_4).C_2^4$ x 5, $C_2\times A_4^2:D_6$ x 4, $C_2\times A_4^2.D_6$ x 3, $C_2\times A_4.D_6^2$, $C_2\times (C_6^2\times A_4).C_2^2$, $C_2^2.C_6^3.C_2^2$, $C_2\times (C_3\times C_6\times A_4).C_2^3$, $C_2\times A_4.C_6^2:C_2.C_2$, $C_2\times C_2^4.C_3^3:C_2.C_2$, $C_2^2\times (C_6^2\times A_4).C_2$, $C_2\times (C_3\times C_6\times A_4).C_2^3$, $C_2\times A_4.(C_2\times D_6:S_3)$, $C_2^2:C_6^2\times S_4$, $C_2\times A_4^2:D_6$, $A_4^2:D_{12}$, $(C_3\times A_4^2):D_4$, $A_4^2:(C_4\times S_3)$, $C_2\times A_4^2:D_6$, $C_6^2.(C_4\times S_4)$, $(C_2^3\times C_6^2):D_6$, $(C_2^2\times C_6^2):D_{12}$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $(C_3\times A_4^2):D_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_6^2):C_6$, $C_3\times A_4^2:D_4$

Order 2592: $C_6^3:D_6$ x 3, $C_2\times A_4^2.C_3^2$ x 2, $(A_4\times C_6^2):S_3$ x 2, $A_4.(C_6^2\times S_3)$, $C_2^2\times (C_3^3\times A_4).C_2$, $C_2\times (C_3^2\times A_4).D_6$, $A_4\times C_3^2:S_4$, $(C_3\times A_4)\wr C_2$, $C_6^2:\SOPlus(4,2)$

Order 2304: $C_6^2.C_2^4.C_2^2$ x 2, $C_6^2.C_2^4.C_2^2$ x 2, $C_2^2\times S_4^2$ x 2, $C_2\wr S_3^2$ x 2, $(C_2^2\times A_4^2):C_4$ x 2, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^5.C_2$, $C_6^2.C_2^6$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^5.C_2$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$, $C_{12}:(C_2^3\times S_4)$, $C_2^4:D_6^2$, $C_2\wr C_6:S_3$, $C_2^6:S_3^2$, $S_4^2:C_4$, $S_4^2:C_2^2$

Order 1728: $A_4^2:D_6$ x 12, $C_6^2:(C_2\times S_4)$ x 10, $A_4:D_6^2$ x 8, $C_2\times C_6^2:S_4$ x 6, $C_2\times C_6^2:S_4$ x 6, $C_2^6:C_3^3$ x 4, $A_4\times C_6^2:C_2^2$ x 4, $C_2\times C_6^2:S_4$ x 3, $C_3:S_4^2$ x 3, $C_2\times C_3^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_2\times C_6^2:D_{12}$ x 3, $C_6^2:(C_4\times D_6)$ x 3, $A_4^2:D_6$ x 2, $C_6\times A_4\times S_4$ x 2, $C_2^3:C_6^3$, $C_2^6:C_3^3$, $S_3\times C_2^3:C_6^2$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_6:D_6\times S_4$, $A_4^2:D_6$, $A_4\wr C_2\times C_6$, $A_4^2:D_6$, $C_6^3:C_2^3$, $A_4\times C_6^2:C_4$, $(C_2\times C_6^2).S_4$, $C_2\times C_6^2.S_4$, $C_6^2:(C_2\times S_4)$, $(C_3\times A_4^2):C_4$, $A_4^2:C_{12}$, $(C_2^3\times C_6^2):C_6$, $(C_6\times D_6):S_4$, $(C_2^2\times C_6^2):C_{12}$

Order 1296: $C_6^3:S_3$ x 3, $C_6^2:S_3^2$ x 3, $C_6^3:C_6$ x 2, $C_6^3:S_3$ x 2, $C_6^3:C_6$, $C_6^2:S_3^2$, $C_2\times C_3^4:D_4$, $C_3^2\times A_4^2$, $C_6^2:S_3^2$, $(C_3^2\times C_6^2):C_4$, $(C_3^3\times C_6).D_4$

Order 1152: $D_6^2.C_2^3$ x 9, $A_4^2:C_2^3$ x 8, $C_2\times S_4^2$ x 6, $C_6:S_4\times D_4$ x 6, $C_2^5:S_3^2$ x 6, $C_2\times C_6:\GL(2,\mathbb{Z}/4)$ x 5, $(C_2^3\times C_6):S_4$ x 4, $D_6^2:D_4$ x 3, $C_2\times C_6^2.(C_2\times D_4)$ x 2, $C_2\times C_6^2.C_2^2:C_4$ x 2, $C_2\times C_3^2.C_2\wr C_4$ x 2, $C_2\times C_3^2.C_2\wr C_4$ x 2, $C_6^2.C_2^4.C_2$ x 2, $C_6^2.(C_4\times D_4)$ x 2, $C_2^3\times C_6:S_4$ x 2, $A_4^2:C_2^3$ x 2, $A_4^2:C_2^3$ x 2, $C_2\times A_4^2:C_4$ x 2, $A_4^2:D_4$ x 2, $A_4^2:D_4$ x 2, $C_2^5.S_3^2$ x 2, $D_6^2.C_2^3$, $C_2\times C_3^2.D_4:D_4$, $(C_2\times D_6:S_3).C_2^3$, $D_6^2:D_4$, $(C_2\times C_6^2).C_2^4$, $C_6^2.C_2^4.C_2$, $C_3^2:C_4.C_2^4.C_2$, $(C_2^3\times C_3:S_3).C_2^3$, $C_6^2.(C_4\times D_4)$, $C_3^2:C_4.(C_4\times D_4)$, $C_3^2.(C_2^5.C_2).C_2$, $C_3^2:C_4.(C_4\times D_4)$, $C_6^2.(C_2^2\times C_4).C_2$, $(C_6\times C_6.D_4).C_2^2$, $C_6^2.(C_2^2\times C_4).C_2$, $C_6\times C_2^3\times S_4$, $C_2^3:D_6^2$, $S_4\wr C_2$, $C_2^3:D_6\times A_4$, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_3\times C_2\wr S_3$, $C_2^2\times C_{12}:S_4$, $C_2^2\times C_{12}:S_4$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $(C_2^3\times C_6):S_4$, $C_2^5:S_3^2$, $C_3:C_2\wr C_6$, $(C_2^3\times C_6):S_4$, $(C_2^2\times C_{12}):S_4$, $(C_2^2\times C_{12}):S_4$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5:C_6^2$, $C_2^5.C_6^2$, $C_6^2.C_2^5$, $C_6^2.C_2^5$, $C_6^2.C_2^5$, $C_2^2\times C_3^2:(C_4:D_4)$, $C_2^2\times C_2^2:(C_3^2:D_4)$, $C_2^2\times (C_6\times S_3):D_4$, $C_2^2\times (C_2^3\times S_3):S_3$, $C_2^2\times D_6^2:C_2$, $C_2^2\times (C_2^3\times C_6):S_3$, $C_2^2\times C_3:C_4\times C_3:D_4$

Order 864: $S_3\times C_6:S_4$ x 17, $C_6^2:S_4$ x 15, $C_6^2:S_4$ x 14, $C_6\times A_4^2$ x 8, $A_4^2:S_3$ x 8, $C_6\times A_4\times D_6$ x 6, $C_6^3:C_2^2$ x 6, $C_6^2:S_4$ x 6, $C_2^2:C_6^3$ x 5, $C_6\wr C_2\times A_4$ x 4, $C_6^2:D_{12}$ x 4, $C_6:S_3\times S_4$ x 3, $C_6^2.S_4$ x 3, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_3^2:(C_4\times S_4)$ x 3, $C_2^5:C_3^3$ x 2, $C_6^2:S_4$ x 2, $C_6:C_{12}\times A_4$ x 2, $C_6^3:C_2^2$, $C_6:D_6^2$, $C_6:D_6\times A_4$, $C_6^2:S_4$, $S_4\times C_6^2$, $A_4^2:S_3$, $A_4^2:C_6$, $A_4^2:S_3$, $A_4^2:C_6$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:(C_4\times S_4)$, $(C_6\times D_6):A_4$

Order 768: $C_2^6:D_6$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2\wr D_6$, $C_2^6:D_6$

Order 648: $C_3^4:C_2^3$ x 4, $C_6^3:C_3$ x 2, $C_3^3:S_4$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$

Order 576: $C_6:\GL(2,\mathbb{Z}/4)$ x 27, $A_4^2:C_2^2$ x 24, $C_2^4:S_3^2$ x 24, $C_2^2\times C_6:S_4$ x 17, $C_2^2:D_6^2$ x 15, $(C_2^2\times C_6):S_4$ x 13, $C_6^2:C_2^4$ x 12, $D_4\times C_3:S_4$ x 11, $C_2^4:S_3^2$ x 10, $C_2^2:D_6^2$ x 9, $C_6^2.C_2^4$ x 9, $C_2^2\times A_4^2$ x 8, $C_2^2\times C_6\times S_4$ x 6, $S_4^2$ x 6, $C_2^4:S_3^2$ x 6, $C_2^2.D_6^2$ x 6, $C_2^2.D_6^2$ x 6, $C_2^2.D_6^2$ x 6, $C_2^2.D_6^2$ x 6, $C_2^2.D_6^2$ x 6, $C_2^4:C_6^2$ x 5, $C_2^4:C_6^2$ x 4, $A_4^2:C_2^2$ x 4, $C_2^4:C_6^2$ x 4, $C_6:D_4\times A_4$ x 4, $C_2^2\times C_6.S_4$ x 4, $C_2\times C_{12}:S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $D_6^2:C_2^2$ x 3, $C_6:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.S_3^2$ x 3, $C_2^4.S_3^2$ x 3, $C_6^2.C_2^4$ x 3, $D_6^2:C_4$ x 3, $(C_2^3\times C_6):C_{12}$ x 3, $C_2^4:S_3^2$ x 3, $C_2^4.S_3^2$ x 3, $(C_2^2\times C_6^2):C_4$ x 3, $A_4^2:C_2^2$ x 2, $\POPlus(4,3)$ x 2, $A_4^2:C_4$ x 2, $C_6^2:C_2^4$ x 2, $C_2^2.D_6^2$ x 2, $A_4^2:C_4$ x 2, $C_6^2.C_2^4$ x 2, $C_3^2:D_4^2$ x 2, $C_2^2.D_6^2$ x 2, $C_2^2.D_6^2$ x 2, $(C_2^2\times C_6^2).C_4$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $C_6^2:\SD_{16}$ x 2, $C_6^2:D_8$ x 2, $D_6^2:C_2^2$ x 2, $C_2^4.S_3^2$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $D_6^2:C_4$ x 2, $C_6^2.C_4^2$ x 2, $C_2^4\times C_6^2$, $C_6^2:C_2^4$, $C_2^2\times D_6^2$, $C_2^2\times A_4\times D_6$, $C_2^2:S_4\times C_6$, $C_6^2:C_2^4$, $C_6^2.C_2^4$, $C_2^4.C_6^2$, $C_2^4:S_3^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_6.C_2^3\times A_4$, $(C_2^2\times C_6).S_4$, $C_3\times C_2^3.S_4$, $C_2\times C_6^2.D_4$, $\GL(2,\mathbb{Z}/4):S_3$, $C_2^4:C_6^2$, $C_2^4:S_3^2$, $C_2^4:S_3^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_6^2:C_4^2$, $(C_2^2\times C_6^2):C_4$, $C_6^2:\OD_{16}$, $C_2^4:(C_6\times S_3)$, $(C_2^2\times C_6):S_4$, $C_2\times C_6^2.D_4$, $C_2^3.\SOPlus(4,2)$, $D_6^2:C_4$, $C_2^4:S_3^2$, $C_3\times C_2^4:C_{12}$, $(C_2^3\times C_6):C_{12}$, $C_2^3.\SOPlus(4,2)$, $C_2^3.\SOPlus(4,2)$

Order 512: $D_4^2:D_4$

Order 432: $C_6^2:D_6$ x 32, $A_4\times C_6^2$ x 17, $C_6^2:D_6$ x 13, $A_4:S_3^2$ x 10, $C_6^2:D_6$ x 10, $C_3:D_6^2$ x 8, $C_6\times S_3\times A_4$ x 8, $C_6^3:C_2$ x 6, $C_6^3:C_2$ x 6, $C_3\times A_4^2$ x 4, $D_6:C_6^2$ x 4, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6:S_3\times A_4$ x 2, $A_4:C_6^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:C_{12}$ x 2, $(C_3^2\times A_4):C_4$ x 2, $C_2\times C_6^3$, $C_6^3:C_2$, $D_6\times C_6^2$, $C_6^2:A_4$, $C_6^3.C_2$, $C_6^2:C_{12}$, $C_3\times C_6.S_4$

Order 288: $D_6\wr C_2$ x 5, $C_2^3\times C_6^2$, $C_2\times A_4^2$

Order 144: $C_2^2\times C_6^2$, $A_4^2$

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 3

Order 64: $C_2^6$

Order 36: $C_6^2$

Order 32: $C_2^2\wr C_2$ x 4, $C_2^5$

Order 18: $C_3\times C_6$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $D_4$ x 8

Order 4: $C_2^2$ x 12

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 41472: $C_6^2:S_4\wr C_2$ ($C_1$)

Order 20736: $C_3^2:S_4^2:C_2^2$ ($C_2$) x 2, $C_3^2:S_4^2:C_4$ ($C_2$) x 2, $C_2\times A_4^2.C_3^2.C_2^3$ ($C_2$), $C_6^2:\POPlus(4,3)$ ($C_2$), $A_4^2:(C_6^2:C_4)$ ($C_2$)

Order 10368: $A_4^2.C_3^2.C_2^3$ ($C_2^2$) x 2, $C_2^5:(C_3^4:C_4)$ ($C_2^2$) x 2, $C_2\times (C_3\times A_4^2).D_6$ ($C_2^2$), $(C_6\times A_4^2).D_6$ ($C_2^2$), $C_2\times A_4^2:S_3^2$ ($C_2^2$)

Order 5184: $C_3^2:S_4^2$ ($D_4$) x 4, $C_2^2\times A_4^2.C_3^2$ ($D_4$), $C_2\times A_4^2.C_3:S_3$ ($D_4$), $(C_3\times A_4^2).D_6$ ($C_2^3$)

Order 2592: $(A_4\times C_6^2):S_3$ ($C_2\times D_4$) x 2, $C_2\times A_4^2.C_3^2$ ($C_2\times D_4$)

Order 1296: $C_3^2\times A_4^2$ ($C_2^2\wr C_2$)

Order 576: $C_2^2\times A_4^2$ ($\SOPlus(4,2)$) x 3, $C_2^4\times C_6^2$ ($\SOPlus(4,2)$)

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

Order 144: $A_4^2$ ($D_6\wr C_2$) x 3, $C_2^2\times C_6^2$ ($D_6\wr C_2$)

Order 64: $C_2^6$ ($C_3^4:D_4$)

Order 36: $C_6^2$ ($S_4\wr C_2$)

Order 32: $C_2^5$ ($C_2\times C_3^4:D_4$)

Order 18: $C_3\times C_6$ ($S_4^2:C_2^2$)

Order 16: $C_2^4$ ($C_6^2:\SOPlus(4,2)$)

Order 9: $C_3^2$ ($S_4^2:D_4$)

Order 4: $C_2^2$ ($A_4^2:\SOPlus(4,2)$)

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

Order 1: $C_1$ ($C_6^2:S_4\wr C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 41472: $C_6^2:S_4\wr C_2$ ($C_1$)

Order 20736: $C_2\times A_4^2.C_3^2.C_2^3$ ($C_2$), $C_3^2:S_4^2:C_2^2$ ($C_2$), $C_3^2:S_4^2:C_4$ ($C_2$), $C_6^2:\POPlus(4,3)$ ($C_2$), $A_4^2:(C_6^2:C_4)$ ($C_2$)

Order 10368: $A_4^2.C_3^2.C_2^3$ ($C_2^2$), $C_2\times (C_3\times A_4^2).D_6$ ($C_2^2$), $(C_6\times A_4^2).D_6$ ($C_2^2$), $C_2\times A_4^2:S_3^2$ ($C_2^2$), $C_2^5:(C_3^4:C_4)$ ($C_2^2$)

Order 5184: $C_2^2\times A_4^2.C_3^2$ ($D_4$), $C_2\times A_4^2.C_3:S_3$ ($D_4$), $(C_3\times A_4^2).D_6$ ($C_2^3$), $C_3^2:S_4^2$ ($D_4$)

Order 2592: $C_2\times A_4^2.C_3^2$ ($C_2\times D_4$), $(A_4\times C_6^2):S_3$ ($C_2\times D_4$)

Order 1296: $C_3^2\times A_4^2$ ($C_2^2\wr C_2$)

Order 576: $C_2^4\times C_6^2$ ($\SOPlus(4,2)$), $C_2^2\times A_4^2$ ($\SOPlus(4,2)$)

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

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

Order 64: $C_2^6$ ($C_3^4:D_4$)

Order 36: $C_6^2$ ($S_4\wr C_2$)

Order 32: $C_2^5$ ($C_2\times C_3^4:D_4$)

Order 18: $C_3\times C_6$ ($S_4^2:C_2^2$)

Order 16: $C_2^4$ ($C_6^2:\SOPlus(4,2)$)

Order 9: $C_3^2$ ($S_4^2:D_4$)

Order 4: $C_2^2$ ($A_4^2:\SOPlus(4,2)$)

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

Order 1: $C_1$ ($C_6^2:S_4\wr C_2$)

Series

Derived series

$C_6^2:S_4\wr C_2$

$\rhd$

$(C_3\times A_4^2).D_6$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_6^2:S_4\wr C_2$

$\rhd$

$C_6^2:\POPlus(4,3)$

$\rhd$

$C_2\times (C_3\times A_4^2).D_6$

$\rhd$

$(C_3\times A_4^2).D_6$

$\rhd$

$(A_4\times C_6^2):S_3$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_6^2:S_4\wr C_2$

$\rhd$

$(C_3\times A_4^2).D_6$

$\rhd$

$C_3^2\times A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

This group is a maximal quotient of 3 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 $189 \times 189$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.