Abstract group 13824.dy: $D_6^2:(C_2^2\times S_4)$

• Label: 13824.dy
• Order: \( 2^{9} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$D_6^2:(C_2^2\times S_4)$
Order:	\(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_6^2.(C_2\times A_4).C_2^6$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \)
Composition factors:	$C_2$ x 9, $C_3$ 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	1087	296	4032	4664	3744	13824
Conjugacy classes	1	35	5	30	49	24	144
Divisions	1	35	5	30	49	24	144
Autjugacy classes	1	24	5	14	32	10	86

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	16	12	16	18	28	8	34	12	144
Irr. rational chars.	16	12	16	18	28	8	34	12	144

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	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:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{4}=d^{6}=e^{6}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,4,3,8)(2,6,5,7)(9,11,13,14,12,10), (1,2,3,5)(4,7,8,6)(9,10,12,11)(13,14), (1,3,6,7)(2,5,8,4), (1,3,7)(2,5,4)(12,13)(15,16)\rangle$
Transitive group:	36T9778	36T9782			more information
Direct product:	$C_2$ $\, \times\, $ $(D_6^2:(C_2\times S_4))$
Semidirect product:	$(D_6^2:D_4)$ $\,\rtimes\,$ $D_6$	$S_3^2$ $\,\rtimes\,$ $(C_2^4:S_4)$	$(D_6^2:S_4)$ $\,\rtimes\,$ $C_2^2$	$(D_6^2:S_4)$ $\,\rtimes\,$ $C_2^2$	all 57
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6^2$ . $(C_2^2\times S_4)$	$C_2^3$ . $(D_6^2:D_6)$	$(C_6:D_6)$ . $(D_4\times S_4)$	$(C_6:S_3)$ . $(C_2^4:S_4)$	all 17
Aut. group:	$\Aut(C_3\times D_6^2)$

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 1318396 subgroups in 35362 conjugacy classes, 135 normal (55 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6^2.(D_4\times S_4)$
Commutator:	$G' \simeq$ $C_6^2:(C_2\times A_4)$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times D_6^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3\times D_4$
Radical:	$R \simeq$ $D_6^2:(C_2^2\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $D_4\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6:D_4$
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

All subgroups of index up to 36 (order at least 384) are shown, as well as all normal subgroups of any index.

Order 13824: $D_6^2:(C_2^2\times S_4)$

Order 6912: $D_6^2:(C_2\times S_4)$ x 8, $C_2^4.S_3^3.C_2$, $C_2\times D_6^2:S_4$, $(C_2^3\times C_6^2):D_{12}$, $C_2\times C_6^2:\GL(2,\mathbb{Z}/4)$, $C_2^5:S_3^3$, $C_6^2:C_2^4.D_6$, $(C_2^3\times D_6^2):C_6$

Order 4608: $(C_2^3\times S_3^2).C_2^4$

Order 3456: $D_6^2:S_4$ x 6, $D_6^2:S_4$ x 6, $C_6^2:\GL(2,\mathbb{Z}/4)$ x 4, $D_6^2:(C_2\times A_4)$ x 4, $D_6^2:S_4$ x 4, $(C_2^2\times C_6^2):D_{12}$ x 4, $C_6^2.(C_4\times S_4)$ x 4, $C_6^2:(C_2^2\times S_4)$ x 2, $(C_2^2\times C_6^2).D_6.C_2$, $C_6^2.(C_2^3\times A_4)$, $C_3\times C_2^4.S_3^2.C_2$, $(C_2^3\times C_6).S_3^2.C_2$, $C_2\times D_6^2:D_6$, $(C_2^3\times C_6^2):D_6$, $C_2^5:C_3^3:C_4$, $C_6\times C_2^4:S_3^2$, $(C_2^3\times C_6^2):C_{12}$, $C_2\times D_6^2:A_4$

Order 2304: $C_6^2.D_4^2$ x 16, $D_6^2.C_2^4$ x 2, $(C_2^3\times S_3^2).C_2^3$ x 2, $(C_2\times D_6:D_6).C_2^3$ x 2, $C_3:S_3.C_2\wr C_2^2.C_2$ x 2, $D_6^2.C_2^4$, $C_6^2.C_2^4.C_2^2$, $D_6^2.C_2^4$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$, $(C_2^3\times S_3^2).C_2^3$, $C_6^2.C_2^4.C_2^2$, $C_2^4:D_6^2$, $C_2^4:D_6^2$

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

Order 1536: $C_2^7:D_6$

Order 1152: $D_6^2:D_4$ x 20, $D_6^2:C_2^3$ x 20, $D_6^2:D_4$ x 20, $D_6^2:D_4$ x 16, $D_6^2:D_4$ x 16, $D_6^2.D_4$ x 16, $C_2^5:S_3^2$ x 15, $D_6^2:D_4$ x 12, $D_6^2:D_4$ x 12, $D_6^2:(C_2\times C_4)$ x 8, $C_6^2.(C_4\times D_4)$ x 8, $D_6^2.D_4$ x 8, $S_3\times C_2^3:S_4$ x 8, $D_6^2:D_4$ x 7, $C_2\times C_3^2.(C_2^5.C_2)$ x 6, $C_2\times (C_2\times S_3^2):C_4.C_2$ x 4, $C_2\times D_6\wr C_2.C_2$ x 4, $D_6^2:C_2^3$ x 4, $C_2^5:S_3^2$ x 3, $C_2\times D_6:D_6.C_2^2$ x 3, $C_2\times C_6^2.C_2^4$ x 3, $C_2\times C_6^2.C_2^4$ x 3, $D_6^2.C_2^3$ x 3, $C_2\times D_6^2.C_2^2$ x 2, $D_6^2.C_2^3$ x 2, $C_2\times (C_2\times C_6^2).C_2^3$ x 2, $C_2\times C_6^2.(C_2^2\times C_4)$ x 2, $C_2\times (C_2\times C_3:S_3).C_4^2$ x 2, $C_2\times C_6^2.C_4^2$ x 2, $D_4\times D_6^2$ x 2, $C_2\times C_6^2.C_2^4$ x 2, $C_2\times C_6^2.C_2^4$ x 2, $C_2\times C_2^2:C_4\times S_3^2$ x 2, $C_2\times C_6^2.(C_2\times D_4)$, $C_2\times C_6^2.(C_2\times D_4)$, $C_2\times C_6^2.(C_2\times D_4)$, $(C_2\times C_6^2).C_2^4$, $C_2\times (C_2\times C_6^2).C_2^3$, $C_2\times C_3:S_3.D_4.C_2^2$, $C_2^2\wr C_3\times S_3$, $(C_2^3\times C_6):S_4$, $C_3\times C_2^2\wr S_3$, $C_2^3:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $C_2^3\times D_6^2$, $C_2\times C_6^2.C_2^4$, $C_2\times C_4:S_3\wr C_2.C_2$, $D_6^2.C_2^3$, $D_6^2.C_2^3$, $C_6^2.C_2^5$, $C_2\times C_2^2\wr C_2\times C_3:S_3$, $C_2\times C_3:S_3.D_4.C_2^2$

Order 864: $S_4\times S_3^2$ x 20, $S_3^2:S_4$ x 12, $C_6^2:S_4$ x 4, $D_6^2:C_6$ x 4, $C_2\times A_4:S_3^2$ x 4, $S_3\times C_6:S_4$ x 4, $C_6\times S_3\times S_4$ x 4, $D_6^2:C_6$ x 4, $C_6^2:D_{12}$ x 4, $C_3^2:C_4\times S_4$ x 4, $C_6^2:(C_2\times A_4)$ x 2, $(C_6\times D_6):A_4$ x 2, $C_6^2:S_4$ x 2, $C_6^2:S_4$ x 2, $C_6:S_3\times S_4$ x 2, $C_6^2:C_6:C_4$ x 2, $(C_6\times D_6):A_4$ x 2, $C_2^5:C_3^3$, $C_2\times C_6^2:C_{12}$, $S_3^3:C_2^2$

Order 768: $C_2\wr D_6$ x 8, $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $C_2^5:D_{12}$, $C_2^5:S_4$, $C_2^6.D_6$, $C_2^7:C_6$, $C_2^6:D_6$, $C_2\wr C_2^2\times D_6$

Order 576: $D_6^2:C_2^2$ x 114, $D_6^2.C_2^2$ x 64, $C_3^2:D_4^2$ x 64, $C_4:D_6^2$ x 58, $D_6^2:C_2^2$ x 48, $D_6^2:C_2^2$ x 48, $D_6^2:C_4$ x 40, $D_6^2:C_2^2$ x 36, $C_2^4:S_3^2$ x 28, $D_6^2:C_2^2$ x 28, $D_6^2:C_2^2$ x 26, $D_6^2:C_4$ x 26, $C_2^4:S_3^2$ x 25, $C_2^2\times D_6^2$ x 24, $D_6^2:C_4$ x 22, $D_6^2:C_2^2$ x 16, $D_6^2.C_2^2$ x 16, $D_6^2.C_2^2$ x 16, $C_6^2.C_2^4$ x 16, $C_2^4:S_3^2$ x 16, $D_6^2:C_2^2$ x 16, $D_6^2:C_2^2$ x 16, $D_6^2:C_4$ x 12, $C_6^2.C_2^4$ x 12, $C_2^4:S_3^2$ x 12, $D_6^2:C_4$ x 12, $(C_2^2\times C_6):S_4$ x 10, $C_2^2:S_4\times C_6$ x 10, $C_6^2.(C_2^2\times C_4)$ x 10, $(C_2^2\times C_6^2):C_4$ x 10, $C_2^2:A_4\times D_6$ x 8, $C_2^2:D_6^2$ x 8, $D_6^2.C_2^2$ x 8, $C_3\times C_2^4:D_6$ x 8, $C_2^4:S_3^2$ x 8, $(C_2\times C_{12}):D_{12}$ x 8, $(C_2\times D_6):D_{12}$ x 8, $C_6^2.C_4^2$ x 8, $C_2^3.\SOPlus(4,2)$ x 8, $D_6^2:C_2^2$ x 7, $C_2^2\times C_6^2:C_4$ x 7, $C_2^2:D_6^2$ x 6, $D_6^2.C_2^2$ x 6, $D_6^2.C_2^2$ x 6, $C_2^2.D_6^2$ x 6, $D_6^2:C_2^2$ x 6, $D_6^2:C_4$ x 6, $C_6^2.(C_2^2\times C_4)$ x 6, $D_6^2:C_4$ x 6, $(C_6\times D_4).D_6$ x 6, $C_2^2.D_6^2$ x 5, $C_2^2:D_6^2$ x 4, $C_6^2:(C_2^2\times C_4)$ x 4, $D_6^2.C_2^2$ x 4, $C_2^4:S_3^2$ x 4, $D_6^2:C_2^2$ x 4, $S_3\times C_2^3:A_4$ x 4, $(C_2^2\times C_6):S_4$ x 4, $C_3\times C_2^3:S_4$ x 4, $C_6^2:C_2^4$ x 3, $C_4:D_6^2$ x 3, $C_6^2.C_2^4$ x 3, $C_6^2.C_2^4$ x 3, $C_6^2.C_2^4$ x 3, $C_6^2.C_2^4$ x 3, $C_2^2.D_6^2$ x 3, $C_2^2.D_6^2$ x 3, $C_2^2.D_6^2$ x 3, $D_6^2:C_4$ x 3, $C_6^2:C_2^4$ x 2, $C_2^4:S_3^2$ x 2, $D_6^2:C_2^2$ x 2, $C_4.D_6^2$ x 2, $C_4:D_6^2$ x 2, $C_4\times D_6^2$ 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, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $(C_2^2\times D_6):C_{12}$ x 2, $C_2^4.S_3^2$ x 2, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $D_6^2:C_2^2$, $C_4.D_6^2$, $C_6^2.(C_2^2\times C_4)$, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $C_2^4:C_6^2$, $(C_2^2\times C_6^2):C_4$, $C_6^2.(C_2^2\times C_4)$, $(C_6\times D_4):C_{12}$, $C_2^4.S_3^2$

Order 512: $C_2^6:D_4$

Order 432: $A_4:S_3^2$ x 16, $C_6^2:D_6$ x 16, $A_4\times S_3^2$ x 13, $A_4:S_3^2$ x 13, $C_6^2:D_6$ x 8, $S_3^3:C_2$ x 8, $C_6\times S_3\times A_4$ x 5, $C_6^2:D_6$ x 5, $(C_3^2\times A_4):C_4$ x 4, $C_6:S_3\times A_4$ x 3, $C_6^2:D_6$ x 2, $A_4:C_6^2$ x 2, $C_2\times S_3^3$ x 2, $S_3^2:D_6$ x 2, $C_6^2:C_{12}$ x 2, $C_6^2:A_4$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$

Order 384: $C_2^5:D_6$ x 51, $C_2\wr C_2^2\times S_3$ x 16, $C_2^2\wr S_3$ x 12, $C_2^4:S_4$ x 9, $C_2\wr C_6$ x 4, $C_2^4:S_4$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5.D_6$ x 4, $C_2\wr S_3$ x 4, $C_{12}:C_2^5$ x 3, $C_2^5:D_6$ x 3, $C_2^4:D_{12}$ x 3, $C_2^5.D_6$ x 3, $C_2^5:D_6$ x 3, $C_2^3:C_4\times D_6$ x 3, $C_2^4:D_{12}$ x 3, $C_2^5:C_{12}$, $D_6\times C_2^5$, $C_2^5:A_4$, $D_{12}:C_2^4$, $C_2^4.S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^6:C_6$, $C_2^5:D_6$, $C_2^5:A_4$, $C_2\wr C_2^2\times C_6$, $(C_6\times D_4):D_4$

Order 288: $C_6^2:D_4$ x 12, $C_6^2:C_2^3$ x 2, $C_2\times D_6^2$ x 2, $C_2^3\times C_6^2$, $C_2\times C_6^2:C_4$

Order 192: $C_2^3:S_4$ x 12, $C_2^3:S_4$

Order 144: $S_3^2:C_2^2$ x 91, $D_6^2$ x 4, $C_6^2:C_4$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_2^2$

Order 96: $C_2^2\times S_4$ x 10, $C_2^2:S_4$ x 2, $C_2^3:A_4$, $D_4\times D_6$

Order 72: $\SOPlus(4,2)$ x 76, $C_6:D_6$ x 2, $C_2\times C_6^2$, $S_3\times D_6$

Order 48: $C_2\times S_4$ x 40, $S_3\times D_4$ x 8, $C_2^2:A_4$

Order 36: $S_3^2$ x 2, $C_6^2$, $C_6:S_3$

Order 32: $C_2^2\times D_4$ x 8, $C_2^5$

Order 27: $C_3^3$

Order 24: $S_4$ x 22, $C_2\times D_6$ x 10

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $C_2\times D_4$ x 91, $C_2^4$

Order 12: $D_6$ x 36

Order 9: $C_3^2$

Order 8: $D_4$ x 76, $C_2^3$ x 73

Order 6: $S_3$ x 16

Order 4: $C_2^2$ x 173

Order 2: $C_2$ x 28

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 384) are shown, as well as all normal subgroups of any index.

Order 13824: $D_6^2:(C_2^2\times S_4)$

Order 6912: $C_2^4.S_3^3.C_2$, $C_2\times D_6^2:S_4$, $(C_2^3\times C_6^2):D_{12}$, $C_2\times C_6^2:\GL(2,\mathbb{Z}/4)$, $C_2^5:S_3^3$, $C_6^2:C_2^4.D_6$, $(C_2^3\times D_6^2):C_6$, $D_6^2:(C_2\times S_4)$

Order 4608: $(C_2^3\times S_3^2).C_2^4$

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

Order 2304: $D_6^2.C_2^4$ x 2, $(C_2^3\times S_3^2).C_2^3$ x 2, $(C_2\times D_6:D_6).C_2^3$ x 2, $C_3:S_3.C_2\wr C_2^2.C_2$ x 2, $C_6^2.D_4^2$ x 2, $D_6^2.C_2^4$, $C_6^2.C_2^4.C_2^2$, $D_6^2.C_2^4$, $C_6^2.C_2^4.C_2^2$, $C_6^2.C_2^4.C_2^2$, $(C_2^3\times S_3^2).C_2^3$, $C_6^2.C_2^4.C_2^2$, $C_2^4:D_6^2$, $C_2^4:D_6^2$

Order 1728: $C_6^2:(C_2\times S_4)$ x 4, $A_4:D_6^2$ x 3, $D_6^2:D_6$ x 3, $C_2\times C_6^2:S_4$ x 2, $D_6^2:A_4$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $D_6^2:A_4$ x 2, $C_6^2:(C_2^2\times A_4)$, $C_2^4:C_3^2\times D_6$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $(C_6\times D_6):S_4$, $C_3\times C_2^4:S_3^2$, $C_6^2:A_4:C_4$, $C_2\times D_6^2:C_6$, $C_2\times C_6^2:D_{12}$, $C_6^2:(C_4\times D_6)$, $D_6^2:D_6$, $(C_2^3\times C_6^2):C_6$, $(C_6\times D_6):S_4$, $C_3\times C_2^4:S_3^2$, $(C_2^2\times C_6^2):C_{12}$

Order 1536: $C_2^7:D_6$

Order 1152: $C_2^5:S_3^2$ x 9, $D_6^2:D_4$ x 7, $C_2\times C_3^2.(C_2^5.C_2)$ x 6, $D_6^2:D_4$ x 6, $D_6^2:D_4$ x 5, $D_6^2:D_4$ x 4, $C_2\times (C_2\times S_3^2):C_4.C_2$ x 4, $D_6^2:D_4$ x 4, $D_6^2:C_2^3$ x 4, $C_2\times D_6\wr C_2.C_2$ x 4, $D_6^2:C_2^3$ x 4, $C_2^5:S_3^2$ x 3, $C_2\times D_6:D_6.C_2^2$ x 3, $C_2\times C_6^2.C_2^4$ x 3, $D_6^2:D_4$ x 2, $C_2\times (C_2\times C_6^2).C_2^3$ x 2, $C_2\times C_6^2.(C_2^2\times C_4)$ x 2, $D_6^2:D_4$ x 2, $D_6^2.D_4$ x 2, $C_6^2.(C_4\times D_4)$ x 2, $C_2\times (C_2\times C_3:S_3).C_4^2$ x 2, $C_2\times C_6^2.C_4^2$ x 2, $S_3\times C_2^3:S_4$ x 2, $D_4\times D_6^2$ x 2, $C_2\times C_6^2.C_2^4$ x 2, $C_2\times C_6^2.C_2^4$ x 2, $C_2\times C_6^2.C_2^4$ x 2, $D_6^2.C_2^3$ x 2, $C_2\times C_2^2:C_4\times S_3^2$ x 2, $C_2\times D_6^2.C_2^2$, $D_6^2.C_2^3$, $C_2\times C_6^2.(C_2\times D_4)$, $C_2\times C_6^2.(C_2\times D_4)$, $C_2\times C_6^2.(C_2\times D_4)$, $(C_2\times C_6^2).C_2^4$, $C_2\times (C_2\times C_6^2).C_2^3$, $C_2\times C_3:S_3.D_4.C_2^2$, $D_6^2:(C_2\times C_4)$, $D_6^2.D_4$, $C_2^2\wr C_3\times S_3$, $(C_2^3\times C_6):S_4$, $C_3\times C_2^2\wr S_3$, $C_2^3:A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_6\times C_2^3:S_4$, $C_2^3\times D_6^2$, $C_2\times C_6^2.C_2^4$, $C_2\times C_4:S_3\wr C_2.C_2$, $D_6^2.C_2^3$, $D_6^2.C_2^3$, $C_6^2.C_2^5$, $C_2\times C_2^2\wr C_2\times C_3:S_3$, $C_2\times C_3:S_3.D_4.C_2^2$

Order 864: $S_4\times S_3^2$ x 6, $D_6^2:C_6$ x 4, $C_2\times A_4:S_3^2$ x 4, $S_3\times C_6:S_4$ x 3, $C_6\times S_3\times S_4$ x 3, $S_3^2:S_4$ x 3, $C_6^2:(C_2\times A_4)$ x 2, $C_6^2:S_4$ x 2, $C_6:S_3\times S_4$ x 2, $C_6^2:C_6:C_4$ x 2, $C_2^5:C_3^3$, $(C_6\times D_6):A_4$, $C_6^2:S_4$, $C_6^2:S_4$, $C_2\times C_6^2:C_{12}$, $S_3^3:C_2^2$, $D_6^2:C_6$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$, $(C_6\times D_6):A_4$

Order 768: $C_2^5:S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $C_2\wr D_6$, $C_2^5:D_{12}$, $C_2^5:S_4$, $C_2^6.D_6$, $C_2^7:C_6$, $C_2^6:D_6$, $C_2\wr C_2^2\times D_6$

Order 576: $D_6^2:C_2^2$ x 54, $C_4:D_6^2$ x 30, $C_2^2\times D_6^2$ x 18, $C_2^4:S_3^2$ x 16, $D_6^2:C_4$ x 12, $D_6^2:C_4$ x 12, $C_2^4:S_3^2$ x 11, $D_6^2:C_2^2$ x 10, $C_2^2:D_6^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2.C_2^2$ x 8, $D_6^2.C_2^2$ x 8, $C_3^2:D_4^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_2^2$ x 8, $D_6^2:C_4$ x 8, $D_6^2:C_4$ x 8, $(C_2^2\times C_6):S_4$ x 7, $C_2^2:S_4\times C_6$ x 7, $D_6^2:C_2^2$ x 7, $C_2^2\times C_6^2:C_4$ x 7, $D_6^2:C_2^2$ x 7, $D_6^2:C_2^2$ x 7, $C_2^2:A_4\times D_6$ x 6, $D_6^2.C_2^2$ x 6, $D_6^2.C_2^2$ x 6, $C_6^2.C_2^4$ x 6, $C_2^2.D_6^2$ x 5, $C_2^2.D_6^2$ x 5, $(C_2^2\times C_6^2):C_4$ x 5, $C_2^2:D_6^2$ x 4, $C_6^2.(C_2^2\times C_4)$ x 4, $C_6^2:(C_2^2\times C_4)$ x 4, $D_6^2.C_2^2$ x 4, $D_6^2:C_4$ x 4, $D_6^2:C_4$ x 4, $C_2^2:D_6^2$ x 3, $C_6^2.C_2^4$ x 3, $C_6^2.C_2^4$ x 3, $C_2^4:S_3^2$ x 3, $D_6^2:C_2^2$ x 3, $D_6^2:C_4$ x 3, $D_6^2:C_4$ x 3, $C_2^4:S_3^2$ x 3, $C_6^2:C_2^4$ x 2, $C_6^2:C_2^4$ x 2, $C_2^4:S_3^2$ x 2, $D_6^2:C_2^2$ x 2, $C_4.D_6^2$ x 2, $C_4:D_6^2$ x 2, $C_4:D_6^2$ x 2, $C_4\times D_6^2$ 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, $D_6^2.C_2^2$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_6^2.C_2^4$ x 2, $C_2^4:S_3^2$ x 2, $C_2^2.D_6^2$ x 2, $C_2^2.D_6^2$ x 2, $C_2^2.D_6^2$ x 2, $D_6^2:C_2^2$ x 2, $C_6^2.(C_2^2\times C_4)$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $S_3\times C_2^3:A_4$ x 2, $C_3\times C_2^4:D_6$ x 2, $(C_2^2\times D_6):C_{12}$ x 2, $C_2^4:S_3^2$ x 2, $C_2^4.S_3^2$ x 2, $(C_2\times C_{12}):D_{12}$ x 2, $(C_6\times D_4).D_6$ x 2, $(C_2\times D_6):D_{12}$ x 2, $C_6^2.C_4^2$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $D_6^2:C_2^2$, $C_4.D_6^2$, $C_6^2.(C_2^2\times C_4)$, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $C_2^4:C_6^2$, $(C_2^2\times C_6^2):C_4$, $C_6^2.(C_2^2\times C_4)$, $(C_2^2\times C_6):S_4$, $C_3\times C_2^3:S_4$, $D_6^2:C_2^2$, $D_6^2:C_2^2$, $(C_6\times D_4):C_{12}$, $C_2^4.S_3^2$

Order 512: $C_2^6:D_4$

Order 432: $A_4\times S_3^2$ x 8, $A_4:S_3^2$ x 8, $C_6\times S_3\times A_4$ x 4, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 4, $C_6:S_3\times A_4$ x 3, $A_4:S_3^2$ x 3, $C_6^2:D_6$ x 3, $C_6^2:D_6$ x 2, $A_4:C_6^2$ x 2, $C_2\times S_3^3$ x 2, $S_3^2:D_6$ x 2, $(C_3^2\times A_4):C_4$ x 2, $C_6^2:A_4$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_6^2:C_{12}$, $S_3^3:C_2$

Order 384: $C_2^5:D_6$ x 21, $C_2^2\wr S_3$ x 8, $C_2^4:S_4$ x 5, $C_2\wr C_2^2\times S_3$ x 4, $C_2^5:D_6$ x 3, $C_2^3:C_4\times D_6$ x 3, $C_2^4:D_{12}$ x 3, $C_{12}:C_2^5$ x 2, $C_2^5:D_6$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2\wr C_6$, $C_2^5:C_{12}$, $C_2^4:S_4$, $C_2^4:D_{12}$, $C_2^5.D_6$, $D_6\times C_2^5$, $C_2^5:A_4$, $D_{12}:C_2^4$, $C_2\wr S_3$, $C_2^4.S_4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^6:C_6$, $C_2^5:D_6$, $C_2^5:A_4$, $C_2\wr C_2^2\times C_6$, $(C_6\times D_4):D_4$

Order 288: $C_6^2:D_4$ x 6, $C_6^2:C_2^3$ x 2, $C_2\times D_6^2$ x 2, $C_2^3\times C_6^2$, $C_2\times C_6^2:C_4$

Order 192: $C_2^3:S_4$ x 3, $C_2^3:S_4$

Order 144: $S_3^2:C_2^2$ x 38, $D_6^2$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 96: $C_2^2\times S_4$ x 4, $C_2^3:A_4$, $C_2^2:S_4$, $D_4\times D_6$

Order 72: $\SOPlus(4,2)$ x 15, $C_6:D_6$ x 2, $C_2\times C_6^2$, $S_3\times D_6$

Order 48: $C_2\times S_4$ x 14, $C_2^2:A_4$, $S_3\times D_4$

Order 36: $C_6^2$, $C_6:S_3$, $S_3^2$

Order 32: $C_2^2\times D_4$ x 5, $C_2^5$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 8, $S_4$ x 8

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

Order 16: $C_2\times D_4$ x 38, $C_2^4$

Order 12: $D_6$ x 17

Order 9: $C_3^2$

Order 8: $C_2^3$ x 28, $D_4$ x 15

Order 6: $S_3$ x 8

Order 4: $C_2^2$ x 76

Order 2: $C_2$ x 17

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 13824: $D_6^2:(C_2^2\times S_4)$ ($C_1$)

Order 6912: $D_6^2:(C_2\times S_4)$ ($C_2$) x 8, $C_2^4.S_3^3.C_2$ ($C_2$), $C_2\times D_6^2:S_4$ ($C_2$), $(C_2^3\times C_6^2):D_{12}$ ($C_2$), $C_2\times C_6^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^5:S_3^3$ ($C_2$), $C_6^2:C_2^4.D_6$ ($C_2$), $(C_2^3\times D_6^2):C_6$ ($C_2$)

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

Order 2304: $D_6^2.C_2^4$ ($S_3$)

Order 1728: $C_6^2:(C_2\times S_4)$ ($C_2^3$) x 4, $D_6^2:A_4$ ($C_2^3$) x 2, $C_6^2:(C_2\times S_4)$ ($C_2^3$) x 2, $C_6^2:(C_2\times S_4)$ ($D_4$) x 2, $C_6^2:A_4:C_4$ ($C_2^3$) x 2, $D_6^2:A_4$ ($C_2^3$) x 2, $(C_2^2\times C_6^2):C_{12}$ ($C_2^3$) x 2, $C_6^2:(C_2^2\times A_4)$ ($C_2^3$), $C_2\times C_6^2:S_4$ ($D_4$), $C_2\times C_6^2:S_4$ ($D_4$)

Order 1152: $D_6^2:D_4$ ($D_6$) x 4, $C_2\times C_6^2.(C_2^2\times C_4)$ ($D_6$), $C_2^3\times D_6^2$ ($D_6$), $C_2\times C_6^2.C_2^4$ ($D_6$)

Order 864: $C_6^2:S_4$ ($C_2\times D_4$) x 2, $C_6^2:S_4$ ($C_2\times D_4$) x 2, $C_2^5:C_3^3$ ($C_2\times D_4$), $C_6^2:(C_2\times A_4)$ ($C_2^4$), $C_6^2:(C_2\times A_4)$ ($C_2\times D_4$)

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

Order 432: $C_6^2:A_4$ ($C_2^2\times D_4$)

Order 288: $C_6^2:D_4$ ($C_2\times S_4$) x 4, $C_2\times D_6^2$ ($C_2\times S_4$) x 2, $C_2^3\times C_6^2$ ($S_3\times D_4$), $C_6^2:C_2^3$ ($C_2^2\times D_6$), $C_6^2:C_2^3$ ($S_3\times D_4$), $C_2\times C_6^2:C_4$ ($C_2\times S_4$)

Order 192: $C_2^3:S_4$ ($\SOPlus(4,2)$)

Order 144: $D_6^2$ ($C_2^2\times S_4$) x 4, $C_6^2:C_4$ ($C_2^2\times S_4$) x 2, $C_2^2\times C_6^2$ ($D_4\times D_6$), $C_6^2:C_2^2$ ($C_2^2\times S_4$)

Order 96: $C_2^2:S_4$ ($S_3^2:C_2^2$) x 2, $C_2^3:A_4$ ($S_3^2:C_2^2$)

Order 72: $C_2\times C_6^2$ ($D_4\times S_4$), $C_6:D_6$ ($C_2^3\times S_4$), $C_6:D_6$ ($D_4\times S_4$), $S_3\times D_6$ ($C_2^3:S_4$)

Order 48: $C_2^2:A_4$ ($C_6^2:D_4$)

Order 36: $S_3^2$ ($C_2^4:S_4$) x 2, $C_6^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_6:S_3$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($S_3^3:C_2$)

Order 18: $C_3\times C_6$ ($C_2\wr D_6$), $C_3:S_3$ ($C_2^5:S_4$), $C_3:S_3$ ($C_2\wr D_6$)

Order 16: $C_2^4$ ($S_3^3:C_2^2$)

Order 9: $C_3^2$ ($C_2^7:D_6$)

Order 8: $C_2^3$ ($D_6^2:D_6$)

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

Order 2: $C_2$ ($C_6^2.(D_4\times S_4)$)

Order 1: $C_1$ ($D_6^2:(C_2^2\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 13824: $D_6^2:(C_2^2\times S_4)$ ($C_1$)

Order 6912: $C_2^4.S_3^3.C_2$ ($C_2$), $C_2\times D_6^2:S_4$ ($C_2$), $(C_2^3\times C_6^2):D_{12}$ ($C_2$), $C_2\times C_6^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^5:S_3^3$ ($C_2$), $C_6^2:C_2^4.D_6$ ($C_2$), $(C_2^3\times D_6^2):C_6$ ($C_2$), $D_6^2:(C_2\times S_4)$ ($C_2$)

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

Order 2304: $D_6^2.C_2^4$ ($S_3$)

Order 1728: $C_6^2:(C_2\times S_4)$ ($C_2^3$) x 2, $C_6^2:(C_2^2\times A_4)$ ($C_2^3$), $C_2\times C_6^2:S_4$ ($D_4$), $C_2\times C_6^2:S_4$ ($D_4$), $D_6^2:A_4$ ($C_2^3$), $C_6^2:(C_2\times S_4)$ ($C_2^3$), $C_6^2:(C_2\times S_4)$ ($D_4$), $C_6^2:A_4:C_4$ ($C_2^3$), $D_6^2:A_4$ ($C_2^3$), $(C_2^2\times C_6^2):C_{12}$ ($C_2^3$)

Order 1152: $D_6^2:D_4$ ($D_6$), $C_2\times C_6^2.(C_2^2\times C_4)$ ($D_6$), $C_2^3\times D_6^2$ ($D_6$), $C_2\times C_6^2.C_2^4$ ($D_6$)

Order 864: $C_2^5:C_3^3$ ($C_2\times D_4$), $C_6^2:(C_2\times A_4)$ ($C_2^4$), $C_6^2:(C_2\times A_4)$ ($C_2\times D_4$), $C_6^2:S_4$ ($C_2\times D_4$), $C_6^2:S_4$ ($C_2\times D_4$)

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

Order 432: $C_6^2:A_4$ ($C_2^2\times D_4$)

Order 288: $C_2\times D_6^2$ ($C_2\times S_4$) x 2, $C_2^3\times C_6^2$ ($S_3\times D_4$), $C_6^2:C_2^3$ ($C_2^2\times D_6$), $C_6^2:C_2^3$ ($S_3\times D_4$), $C_2\times C_6^2:C_4$ ($C_2\times S_4$), $C_6^2:D_4$ ($C_2\times S_4$)

Order 192: $C_2^3:S_4$ ($\SOPlus(4,2)$)

Order 144: $D_6^2$ ($C_2^2\times S_4$) x 2, $C_2^2\times C_6^2$ ($D_4\times D_6$), $C_6^2:C_2^2$ ($C_2^2\times S_4$), $C_6^2:C_4$ ($C_2^2\times S_4$)

Order 96: $C_2^3:A_4$ ($S_3^2:C_2^2$), $C_2^2:S_4$ ($S_3^2:C_2^2$)

Order 72: $C_2\times C_6^2$ ($D_4\times S_4$), $C_6:D_6$ ($C_2^3\times S_4$), $C_6:D_6$ ($D_4\times S_4$), $S_3\times D_6$ ($C_2^3:S_4$)

Order 48: $C_2^2:A_4$ ($C_6^2:D_4$)

Order 36: $C_6^2$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_6:S_3$ ($C_2^4:S_4$), $S_3^2$ ($C_2^4:S_4$)

Order 32: $C_2^5$ ($S_3^3:C_2$)

Order 18: $C_3\times C_6$ ($C_2\wr D_6$), $C_3:S_3$ ($C_2^5:S_4$), $C_3:S_3$ ($C_2\wr D_6$)

Order 16: $C_2^4$ ($S_3^3:C_2^2$)

Order 9: $C_3^2$ ($C_2^7:D_6$)

Order 8: $C_2^3$ ($D_6^2:D_6$)

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

Order 2: $C_2$ ($C_6^2.(D_4\times S_4)$)

Order 1: $C_1$ ($D_6^2:(C_2^2\times S_4)$)

Series

Derived series

$D_6^2:(C_2^2\times S_4)$

$\rhd$

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

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$D_6^2:(C_2^2\times S_4)$

$\rhd$

$D_6^2:(C_2\times S_4)$

$\rhd$

$D_6^2:S_4$

$\rhd$

$D_6^2:A_4$

$\rhd$

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

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$D_6^2:(C_2^2\times S_4)$

$\rhd$

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

$\rhd$

$C_6^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 6 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 $144 \times 144$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.