Abstract group 6912.dm: $(C_2^3\times D_6^2):C_6$

• Label: 6912.dm
• Order: \( 2^{8} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$(C_2^3\times D_6^2):C_6$
Order:	\(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_5\wr C_2^2$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \)
Composition factors:	$C_2$ x 8, $C_3$ 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	559	296	720	3896	1440	6912
Conjugacy classes	1	29	8	6	58	6	108
Divisions	1	29	5	6	45	4	90
Autjugacy classes	1	15	5	3	23	2	49

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	24	6	8	24	18	0	24	4	108
Irr. rational chars.	8	10	8	10	18	8	24	4	90

Minimal presentations

Permutation degree:	$16$
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:	${\langle a, b, c, d, e, f, g \mid a^{6}=b^{4}=c^{6}=d^{6}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,3)(4,5)(7,8)(10,12,15)(11,14,13), (1,3,2)(7,8)(9,10)(12,15), (1,4)(2,5,3,6)(9,11)(10,13,12,14,15,16)\rangle$
Transitive group:	36T6908	36T6911			more information
Direct product:	$C_2$ $\, \times\, $ $(D_6^2:(C_2\times A_4))$
Semidirect product:	$(D_6^2:D_4)$ $\,\rtimes\,$ $C_6$	$S_3^2$ $\,\rtimes\,$ $(C_2^4:A_4)$	$C_2^5$ $\,\rtimes\,$ $(S_3^2:C_6)$	$(D_6^2:A_4)$ $\,\rtimes\,$ $C_2^2$	all 33
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6^2$ . $(C_2^2\times A_4)$	$C_2^3$ . $(D_6^2:C_6)$	$C_6^2$ . $(C_2^5:C_6)$	$(C_6:D_6)$ . $(D_4\times A_4)$	all 15

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 493596 subgroups in 22226 conjugacy classes, 73 normal (41 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_6^2:(C_2\times A_4)$
Commutator:	$G' \simeq$ $C_6^2:C_2^3$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times D_6^2:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_3\times D_4$
Radical:	$R \simeq$ $(C_2^3\times D_6^2):C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $D_4\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
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 6912: $(C_2^3\times D_6^2):C_6$

Order 3456: $D_6^2:(C_2\times A_4)$ x 4, $C_6^2.(C_2^3\times A_4)$, $(C_2^3\times C_6^2):C_{12}$, $C_2\times D_6^2:A_4$

Order 2304: $D_6^2.C_2^4$

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

Order 1152: $D_6^2:D_4$ x 8, $C_2\times C_3^2.(C_2^5.C_2)$ x 2, $D_6^2:D_4$ x 2, $C_2\times C_6^2.(C_2^2\times C_4)$, $C_2^2\wr C_3\times S_3$, $C_2^3:A_4\times D_6$, $C_2^3\times D_6^2$, $C_2\times C_6^2.C_2^4$

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

Order 768: $C_2^7:C_6$

Order 576: $D_6^2:C_2^2$ x 32, $C_2^2\times D_6^2$ x 25, $D_6^2:C_4$ x 16, $C_2^2:A_4\times D_6$ x 8, $C_2^4:S_3^2$ x 6, $D_6^2:C_4$ x 4, $(C_2^2\times C_6^2):C_4$ x 4, $S_3\times C_2^3:A_4$ x 4, $D_6^2:C_2^2$ x 2, $C_2^2\times C_6^2:C_4$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^4:C_6^2$, $C_2^2:D_6^2$, $C_6^2.C_2^4$, $C_2^4:C_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$

Order 432: $A_4\times S_3^2$ x 14, $C_6\times S_3\times A_4$ x 6, $C_6:S_3\times A_4$ x 3, $C_6^2:C_{12}$ x 2, $C_6^2:A_4$, $C_6\times \SOPlus(4,2)$

Order 384: $C_2\wr C_6$ x 4, $C_2^5:C_{12}$, $D_6\times C_2^5$, $C_2^5:A_4$, $C_2^5:D_6$, $C_2^5:A_4$

Order 288: $C_2\times D_6^2$ x 429, $D_6\wr C_2$ x 64, $C_6^2.D_4$ x 32, $C_6^2:D_4$ x 24, $C_6^2:C_2^3$ x 22, $C_2\times C_6^2:C_4$ x 16, $D_6^2:C_2$ x 14, $C_6^2:C_2^3$ x 14, $C_6^2:D_4$ x 8, $S_3\times C_2^2:A_4$ x 8, $C_2^3.S_3^2$ x 6, $C_2^5:C_3^2$ x 6, $C_2\times A_4\times D_6$ x 6, $C_6^2:D_4$ x 4, $(C_2\times D_6):A_4$ x 4, $C_3\times C_2^3:A_4$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2:D_4$, $C_2.D_6^2$, $C_6^2.D_4$, $C_6^2.D_4$, $C_2^3\times C_6^2$, $C_6^2:D_4$

Order 256: $C_2^5:D_4$

Order 216: $C_6^2:C_6$ x 12, $C_6^2:C_6$ x 6, $S_3^2:C_6$ x 4, $C_6^2:C_6$ x 3, $C_6\times S_3^2$ x 2, $C_2\times C_3^2:C_{12}$

Order 192: $D_6\times C_2^4$ x 47, $C_2^4:D_6$ x 8, $C_2^2\wr C_3$ x 6, $C_2^4:A_4$ x 5, $C_2^4:C_{12}$ x 2, $C_2^5\times C_6$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^5:C_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $D_6^2$ x 1693, $C_6^2:C_2^2$ x 211, $C_6^2:C_2^2$ x 114, $S_3^2:C_2^2$ x 54, $A_4\times D_6$ x 44, $S_3^2:C_4$ x 32, $D_6:D_6$ x 24, $C_6^2:C_4$ x 16, $C_6^2:C_4$ x 12, $C_6^2:C_2^2$ x 8, $C_6:D_{12}$ x 8, $C_2^2\times C_6^2$ x 7, $C_2^2.S_3^2$ x 7, $C_2^2:C_6^2$ x 6, $C_6^2:C_4$ x 4, $C_6.D_{12}$ x 4, $C_2^4:C_3^2$ x 4, $C_6^2:C_4$

Order 128: $C_2^3\wr C_2$ x 8, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $C_2^7$

Order 108: $C_3\times S_3^2$ x 6, $C_3^2\times A_4$ x 3, $C_3^2\times D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$

Order 96: $C_2^3\times D_6$ x 889, $C_2^4\times C_6$ x 24, $D_4\times D_6$ x 16, $C_2^3:A_4$ x 9, $C_2^2:D_{12}$ x 8, $D_6.D_4$ x 8, $D_6:D_4$ x 8, $C_2^3:A_4$ x 4, $C_2^3\times A_4$ x 4, $D_4\times A_4$ x 4, $C_2^4:C_6$ x 4, $C_2^4:S_3$ x 4, $C_2^3:D_6$ x 2, $C_2^2.D_{12}$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^3:C_{12}$, $C_2^3.D_6$

Order 72: $S_3\times D_6$ x 1354, $C_6\times D_6$ x 426, $C_6:D_6$ x 224, $S_3\times A_4$ x 40, $C_2\times C_6^2$ x 31, $C_6\times A_4$ x 28, $\SOPlus(4,2)$ x 24, $C_2\times C_3^2:C_4$ x 14, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6.D_6$ x 6, $C_6:C_{12}$ x 4

Order 64: $C_2^3:D_4$ x 40, $C_2^6$ x 27, $C_2^4:C_4$ x 20, $C_2^4:C_4$ x 7, $D_4\times C_2^3$ x 3

Order 54: $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 2, $C_3^2\times C_6$

Order 48: $C_2^2\times D_6$ x 3863, $C_2^3\times C_6$ x 229, $S_3\times D_4$ x 32, $C_2^2\times A_4$ x 29, $C_6:D_4$ x 16, $C_6\times D_4$ x 9, $C_2\times D_{12}$ x 8, $C_4\times D_6$ x 8, $D_6:C_4$ x 8, $C_2^2:C_{12}$ x 4, $C_6.D_4$ x 4, $C_2^2:A_4$ x 3, $C_4\times A_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 184, $S_3^2$ x 184, $C_6:S_3$ x 90, $C_6^2$ x 33, $C_3\times A_4$ x 16, $C_3^2:C_4$ x 4, $C_3:C_{12}$ x 2

Order 32: $C_2^5$ x 466, $C_2^2\wr C_2$ x 76, $C_2^3:C_4$ x 56, $C_2^2\times D_4$ x 40, $C_2^3\times C_4$ x 3

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 3728, $C_2^2\times C_6$ x 512, $C_2\times A_4$ x 41, $C_3:D_4$ x 16, $C_3\times D_4$ x 12, $D_{12}$ x 8, $C_4\times S_3$ x 8, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 4

Order 18: $C_3\times S_3$ x 26, $C_3\times C_6$ x 15, $C_3:S_3$ x 8

Order 16: $C_2^4$ x 2025, $C_2\times D_4$ x 94, $C_2^2:C_4$ x 56, $C_2^2\times C_4$ x 20

Order 12: $D_6$ x 778, $C_2\times C_6$ x 267, $A_4$ x 11, $C_{12}$ x 4, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 2033, $D_4$ x 40, $C_2\times C_4$ x 24

Order 6: $C_6$ x 45, $S_3$ x 34

Order 4: $C_2^2$ x 475, $C_4$ x 6

Order 3: $C_3$ x 5

Order 2: $C_2$ x 29

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6912: $(C_2^3\times D_6^2):C_6$

Order 3456: $C_6^2.(C_2^3\times A_4)$, $(C_2^3\times C_6^2):C_{12}$, $C_2\times D_6^2:A_4$, $D_6^2:(C_2\times A_4)$

Order 2304: $D_6^2.C_2^4$

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

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

Order 864: $D_6^2:C_6$ x 3, $C_6^2:(C_2\times A_4)$ x 2, $C_2^5:C_3^3$, $(C_6\times D_6):A_4$, $C_2\times C_6^2:C_{12}$, $D_6^2:C_6$, $(C_6\times D_6):A_4$

Order 768: $C_2^7:C_6$

Order 576: $C_2^2\times D_6^2$ x 12, $D_6^2:C_2^2$ x 9, $C_2^2:A_4\times D_6$ x 6, $C_2^4:S_3^2$ x 4, $D_6^2:C_4$ x 4, $D_6^2:C_4$ x 3, $D_6^2:C_2^2$ x 2, $C_2^2\times C_6^2:C_4$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_2^4:(C_6\times S_3)$ x 2, $S_3\times C_2^3:A_4$ x 2, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^4:C_6^2$, $C_2^2:D_6^2$, $C_6^2.C_2^4$, $C_2^4:C_6^2$, $C_2^2.D_6^2$, $C_2^2.D_6^2$

Order 432: $A_4\times S_3^2$ x 6, $C_6\times S_3\times A_4$ x 3, $C_6:S_3\times A_4$ x 2, $C_6^2:A_4$, $C_6\times \SOPlus(4,2)$, $C_6^2:C_{12}$

Order 384: $C_2\wr C_6$, $C_2^5:C_{12}$, $D_6\times C_2^5$, $C_2^5:A_4$, $C_2^5:D_6$, $C_2^5:A_4$

Order 288: $C_2\times D_6^2$ x 85, $C_6^2.D_4$ x 10, $C_6^2:C_2^3$ x 10, $C_6^2:D_4$ x 9, $D_6^2:C_2$ x 8, $C_2\times C_6^2:C_4$ x 8, $C_6^2:C_2^3$ x 7, $D_6\wr C_2$ x 6, $S_3\times C_2^2:A_4$ x 6, $C_2^5:C_3^2$ x 5, $C_2^3.S_3^2$ x 4, $C_6^2:D_4$ x 3, $C_2\times A_4\times D_6$ x 3, $C_6^2:D_4$ x 2, $(C_2\times D_6):A_4$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2:D_4$, $C_2.D_6^2$, $C_6^2.D_4$, $C_3\times C_2^3:A_4$, $C_6^2.D_4$, $C_2^3\times C_6^2$, $C_6^2:D_4$

Order 256: $C_2^5:D_4$

Order 216: $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 3, $C_6^2:C_6$ x 2, $C_6\times S_3^2$ x 2, $C_2\times C_3^2:C_{12}$, $S_3^2:C_6$

Order 192: $D_6\times C_2^4$ x 17, $C_2^2\wr C_3$ x 5, $C_2^4:D_6$ x 4, $C_2^4:A_4$ x 4, $C_2^4:C_{12}$, $C_2^5\times C_6$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^5:C_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $D_6^2$ x 221, $C_6^2:C_2^2$ x 48, $C_6^2:C_2^2$ x 34, $A_4\times D_6$ x 16, $S_3^2:C_2^2$ x 11, $C_6^2:C_4$ x 8, $D_6:D_6$ x 8, $C_2^2.S_3^2$ x 6, $C_2^2\times C_6^2$ x 5, $C_2^4:C_3^2$ x 4, $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $C_6^2:C_4$ x 4, $S_3^2:C_4$ x 4, $C_2^2:C_6^2$ x 3, $C_6^2:C_4$ x 2, $C_6.D_{12}$ x 2, $C_6^2:C_4$

Order 128: $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $C_2^3\wr C_2$ x 2, $C_2^7$

Order 108: $C_3\times S_3^2$ x 4, $C_3^2\times D_6$ x 2, $C_3^2\times A_4$ x 2, $C_3^2:D_6$, $C_3^2:C_{12}$

Order 96: $C_2^3\times D_6$ x 138, $C_2^4\times C_6$ x 9, $D_4\times D_6$ x 8, $C_2^3:A_4$ x 7, $D_6.D_4$ x 4, $C_2^2:D_{12}$ x 3, $C_2^3\times A_4$ x 3, $D_6:D_4$ x 3, $C_2^3:A_4$ x 2, $C_2^3:D_6$ x 2, $C_2^4:C_6$ x 2, $C_2^4:S_3$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $D_4\times A_4$, $C_2^3:C_{12}$, $C_2^3:C_{12}$, $C_2^3.D_6$, $C_2^2.D_{12}$

Order 72: $S_3\times D_6$ x 160, $C_6:D_6$ x 85, $C_6\times D_6$ x 55, $C_2\times C_6^2$ x 15, $S_3\times A_4$ x 14, $C_6\times A_4$ x 11, $C_2\times C_3^2:C_4$ x 8, $C_6.D_6$ x 4, $\SOPlus(4,2)$ x 3, $C_6:C_{12}$ x 3, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2

Order 64: $C_2^3:D_4$ x 15, $C_2^6$ x 14, $C_2^4:C_4$ x 6, $C_2^4:C_4$ x 6, $D_4\times C_2^3$ x 3

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

Order 48: $C_2^2\times D_6$ x 452, $C_2^3\times C_6$ x 44, $C_2^2\times A_4$ x 14, $C_6:D_4$ x 8, $S_3\times D_4$ x 8, $C_4\times D_6$ x 6, $C_6\times D_4$ x 5, $C_2\times D_{12}$ x 4, $C_2^2:A_4$ x 3, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $D_6:C_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_4\times A_4$

Order 36: $C_6:S_3$ x 38, $C_6\times S_3$ x 36, $S_3^2$ x 30, $C_6^2$ x 17, $C_3\times A_4$ x 8, $C_3^2:C_4$ x 2, $C_3:C_{12}$

Order 32: $C_2^5$ x 103, $C_2^3:C_4$ x 24, $C_2^2\times D_4$ x 19, $C_2^2\wr C_2$ x 11, $C_2^3\times C_4$ x 3

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 467, $C_2^2\times C_6$ x 85, $C_2\times A_4$ x 16, $C_2\times C_{12}$ x 4, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $C_6:C_4$ x 3, $C_3\times D_4$ x 3, $D_{12}$ x 2

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 11, $C_3:S_3$ x 6

Order 16: $C_2^4$ x 308, $C_2\times D_4$ x 27, $C_2^2\times C_4$ x 15, $C_2^2:C_4$ x 12

Order 12: $D_6$ x 142, $C_2\times C_6$ x 65, $A_4$ x 6, $C_{12}$ x 2, $C_3:C_4$

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 313, $C_2\times C_4$ x 15, $D_4$ x 7

Order 6: $C_6$ x 23, $S_3$ x 16

Order 4: $C_2^2$ x 109, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 15

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6912: $(C_2^3\times D_6^2):C_6$ ($C_1$)

Order 3456: $D_6^2:(C_2\times A_4)$ ($C_2$) x 4, $C_6^2.(C_2^3\times A_4)$ ($C_2$), $(C_2^3\times C_6^2):C_{12}$ ($C_2$), $C_2\times D_6^2:A_4$ ($C_2$)

Order 2304: $D_6^2.C_2^4$ ($C_3$)

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

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

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

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

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

Order 288: $C_6^2:D_4$ ($C_2\times A_4$) x 4, $C_2\times D_6^2$ ($C_2\times A_4$) x 2, $C_2^3\times C_6^2$ ($C_3\times D_4$), $C_6^2:C_2^3$ ($C_2^2\times C_6$), $C_6^2:C_2^3$ ($C_3\times D_4$), $C_2\times C_6^2:C_4$ ($C_2\times A_4$)

Order 144: $D_6^2$ ($C_2^2\times A_4$) x 4, $C_6^2:C_4$ ($C_2^2\times A_4$) x 2, $C_2^2\times C_6^2$ ($C_6\times D_4$), $C_6^2:C_2^2$ ($C_2^2\times A_4$)

Order 96: $C_2^3:A_4$ ($\SOPlus(4,2)$)

Order 72: $C_2\times C_6^2$ ($D_4\times A_4$), $C_6:D_6$ ($C_2^3\times A_4$), $C_6:D_6$ ($D_4\times A_4$), $S_3\times D_6$ ($C_2^3:A_4$)

Order 48: $C_2^2:A_4$ ($S_3^2:C_2^2$)

Order 36: $S_3^2$ ($C_2^4:A_4$) x 2, $C_6^2$ ($C_2^5:C_6$), $C_6:S_3$ ($C_2^4:A_4$)

Order 32: $C_2^5$ ($S_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_2\wr C_6$), $C_3:S_3$ ($C_2\wr C_6$), $C_3:S_3$ ($C_2^5:A_4$)

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

Order 9: $C_3^2$ ($C_2^7:C_6$)

Order 8: $C_2^3$ ($D_6^2:C_6$)

Order 4: $C_2^2$ ($C_2\times D_6^2:C_6$)

Order 2: $C_2$ ($D_6^2:(C_2\times A_4)$)

Order 1: $C_1$ ($(C_2^3\times D_6^2):C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6912: $(C_2^3\times D_6^2):C_6$ ($C_1$)

Order 3456: $C_6^2.(C_2^3\times A_4)$ ($C_2$), $(C_2^3\times C_6^2):C_{12}$ ($C_2$), $C_2\times D_6^2:A_4$ ($C_2$), $D_6^2:(C_2\times A_4)$ ($C_2$)

Order 2304: $D_6^2.C_2^4$ ($C_3$)

Order 1728: $C_6^2:(C_2^2\times A_4)$ ($C_2^2$), $D_6^2:A_4$ ($C_2^2$), $D_6^2:A_4$ ($C_2^2$), $(C_2^2\times C_6^2):C_{12}$ ($C_2^2$)

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

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

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

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

Order 288: $C_2\times D_6^2$ ($C_2\times A_4$) x 2, $C_2^3\times C_6^2$ ($C_3\times D_4$), $C_6^2:C_2^3$ ($C_2^2\times C_6$), $C_6^2:C_2^3$ ($C_3\times D_4$), $C_2\times C_6^2:C_4$ ($C_2\times A_4$), $C_6^2:D_4$ ($C_2\times A_4$)

Order 144: $D_6^2$ ($C_2^2\times A_4$) x 2, $C_2^2\times C_6^2$ ($C_6\times D_4$), $C_6^2:C_2^2$ ($C_2^2\times A_4$), $C_6^2:C_4$ ($C_2^2\times A_4$)

Order 96: $C_2^3:A_4$ ($\SOPlus(4,2)$)

Order 72: $C_2\times C_6^2$ ($D_4\times A_4$), $C_6:D_6$ ($C_2^3\times A_4$), $C_6:D_6$ ($D_4\times A_4$), $S_3\times D_6$ ($C_2^3:A_4$)

Order 48: $C_2^2:A_4$ ($S_3^2:C_2^2$)

Order 36: $C_6^2$ ($C_2^5:C_6$), $C_6:S_3$ ($C_2^4:A_4$), $S_3^2$ ($C_2^4:A_4$)

Order 32: $C_2^5$ ($S_3^2:C_6$)

Order 18: $C_3\times C_6$ ($C_2\wr C_6$), $C_3:S_3$ ($C_2\wr C_6$), $C_3:S_3$ ($C_2^5:A_4$)

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

Order 9: $C_3^2$ ($C_2^7:C_6$)

Order 8: $C_2^3$ ($D_6^2:C_6$)

Order 4: $C_2^2$ ($C_2\times D_6^2:C_6$)

Order 2: $C_2$ ($D_6^2:(C_2\times A_4)$)

Order 1: $C_1$ ($(C_2^3\times D_6^2):C_6$)

Series

Derived series

$(C_2^3\times D_6^2):C_6$

$\rhd$

$C_6^2:C_2^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$(C_2^3\times D_6^2):C_6$

$\rhd$

$C_6^2.(C_2^3\times A_4)$

$\rhd$

$D_6^2:A_4$

$\rhd$

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

$\rhd$

$C_6^2:C_2^3$

$\rhd$

$C_6:D_6$

$\rhd$

$C_6^2$

$\rhd$

$C_3:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$(C_2^3\times D_6^2):C_6$

$\rhd$

$C_6^2:C_2^3$

$\rhd$

$C_2^2\times C_6^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.