Abstract group 6912.fa: $(C_2^3\times C_6^2):D_{12}$

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

Group information

Description:	$(C_2^3\times C_6^2):D_{12}$
Order:	\(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^2.(C_2\times A_4).C_2^6.C_2$, 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	607	296	1440	1688	2880	6912
Conjugacy classes	1	19	5	16	27	16	84
Divisions	1	19	5	16	27	14	82
Autjugacy classes	1	12	3	4	12	3	35

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	8	10	8	8	18	4	24	4	84
Irr. rational chars.	8	6	8	10	18	4	24	4	82

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$806400$

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:	${\langle a, b, c, d, e, f, g \mid b^{12}=c^{6}=d^{6}=e^{2}=f^{2}=g^{2}=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16), (1,3,5,7,8,2)(4,6)(9,10,11,12)(13,14)(15,16), (5,8)(6,7)(9,11,14)(10,12)(15,16)\rangle$
Transitive group:	36T6965	36T6966			more information
Direct product:	$C_2$ $\, \times\, $ $((C_2^2\times C_6^2):D_{12})$
Semidirect product:	$(C_2^5:C_3^3)$ $\,\rtimes\,$ $D_4$	$(C_2^3\times C_6^2)$ $\,\rtimes\,$ $D_{12}$	$(C_6^2:C_2^3)$ $\,\rtimes\,$ $D_{12}$	$C_2^5$ $\,\rtimes\,$ $(C_3^2:D_{12})$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2:C_2^4)$ . $D_6$	$C_6^2$ . $(C_2^3:D_{12})$	$C_2^3$ . $(C_6^2:D_{12})$	$(C_6:D_6)$ . $(C_4:S_4)$	all 13

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

Homology

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

Subgroups

There are 189952 subgroups in 6042 conjugacy classes, 48 normal (32 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_2^2\times C_6^2):D_{12}$
Commutator:	$G' \simeq$ $C_6^2:(C_2\times A_4)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times C_6^2:D_{12}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_{12}$
Radical:	$R \simeq$ $(C_2^3\times C_6^2):D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_4:S_4$
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: $(C_2^3\times C_6^2):D_{12}$

Order 3456: $(C_2^2\times C_6^2):D_{12}$ x 4, $C_6^2:(C_2^2\times S_4)$ x 2, $(C_2^3\times C_6^2):C_{12}$

Order 2304: $C_2\times (C_2\times D_6:D_6).C_2^2$

Order 1728: $C_6^2:(C_2\times S_4)$ x 6, $C_2\times C_6^2:S_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $C_6^2:(C_2^2\times A_4)$, $C_2\times C_6^2:D_{12}$

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

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

Order 768: $C_2^5:D_{12}$

Order 576: $C_2^4:S_3^2$ x 20, $D_6^2:C_4$ x 12, $D_6^2:C_2^2$ x 10, $C_2^4:S_3^2$ x 8, $C_2^2.D_6^2$ x 6, $C_2^2:S_4\times C_6$ x 4, $C_2^2:D_6^2$ x 4, $C_2^2.D_6^2$ x 4, $(C_2^2\times C_6^2):C_4$ x 4, $D_6^2:C_4$ x 4, $C_2^3.\SOPlus(4,2)$ x 4, $C_2^2:A_4\times D_6$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_2^2\times C_6^2:C_4$ 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, $C_6^2:C_2^4$

Order 432: $A_4:S_3^2$ x 14, $C_6^2:D_6$ x 6, $C_6:S_3\times A_4$ x 3, $C_6^2:C_{12}$ x 2, $C_6^2:A_4$, $C_2\times C_3^2:D_{12}$

Order 384: $C_2^4:D_{12}$ x 4, $C_2^2\wr S_3$ x 2, $C_2^5:D_6$ x 2, $C_2^5:C_{12}$

Order 288: $D_6^2:C_2$ x 60, $C_6^2:D_4$ x 48, $C_2^3.S_3^2$ x 28, $C_6^2:D_4$ x 16, $D_6\wr C_2$ x 16, $C_6^2.D_4$ x 16, $C_6^2:D_4$ x 16, $C_6^2.D_4$ x 16, $C_2\times C_6^2:C_4$ x 15, $C_6^2:C_2^3$ x 14, $C_6^2.D_4$ x 8, $C_3\times C_2^2:S_4$ x 8, $C_6^2:D_4$ x 6, $C_6^2.D_4$ x 6, $D_6\times S_4$ x 6, $C_6^2:D_4$ x 6, $C_2.D_6^2$ x 4, $C_6^2.D_4$ x 4, $C_2^5:C_3^2$ x 4, $S_3\times C_2^2:A_4$ x 4, $(C_2\times C_6):S_4$ x 4, $C_6^2.D_4$ x 2, $C_2\times D_6^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3\times C_6^2$

Order 256: $C_2^5:D_4$

Order 216: $C_3^2:S_4$ x 12, $C_6^2:C_6$ x 6, $C_3^2:D_{12}$ x 4, $C_6^2:C_6$ x 3, $C_6:S_3^2$ x 2, $C_2\times C_3^2:C_{12}$

Order 192: $C_2^4:D_6$ x 32, $C_2^3:S_4$ x 10, $C_{12}:C_2^4$ x 6, $C_2^4:D_6$ x 6, $C_2^3:D_{12}$ x 6, $C_2^4.D_6$ x 6, $C_2^4:C_{12}$ x 2, $D_6\times C_2^4$ x 2, $C_2^5:C_6$ x 2, $C_2^4:D_6$ x 2, $C_2^2\wr C_3$, $C_2^3:D_{12}$

Order 144: $D_6:D_6$ x 112, $C_6^2:C_2^2$ x 83, $C_6^2:C_2^2$ x 48, $C_6:D_{12}$ x 48, $C_2^2.S_3^2$ x 30, $C_6.D_{12}$ x 28, $C_6^2:C_4$ x 24, $S_3\times S_4$ x 24, $D_6^2$ x 22, $S_3^2:C_4$ x 16, $C_6^2:C_4$ x 14, $C_6\times S_4$ x 12, $S_3^2:C_2^2$ x 12, $C_6^2:C_4$ x 10, $A_4\times D_6$ x 8, $C_2.\SOPlus(4,2)$ x 8, $C_2^2\times C_6^2$ x 7, $C_6:S_4$ x 6, $C_6^2:C_4$ x 6, $C_2^4:C_3^2$ x 4, $C_6^2:C_2^2$ x 2

Order 128: $C_2^4:D_4$ x 8, $C_2^4:D_4$ x 2, $C_2^5:C_4$ x 2, $C_2^3.C_4^2$, $C_2^4:D_4$, $C_2^5:C_4$

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

Order 96: $D_4\times D_6$ x 96, $C_2^2:D_{12}$ x 48, $D_6.D_4$ x 48, $D_6:D_4$ x 48, $C_2^3\times D_6$ x 38, $C_2^4:C_6$ x 16, $C_2^4:S_3$ x 16, $C_2^3:D_6$ x 12, $C_2^2.D_{12}$ x 12, $C_2^2:S_4$ x 8, $C_{12}:C_2^3$ 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 4, $C_4:S_4$ x 4, $C_2^4\times C_6$ x 2, $C_2^3:C_{12}$

Order 72: $C_6:D_6$ x 152, $S_3\times D_6$ x 50, $C_6\wr C_2$ x 48, $C_3:D_{12}$ x 48, $C_6.D_6$ x 28, $C_2\times C_6^2$ x 25, $C_3\times S_4$ x 24, $C_6:C_{12}$ x 24, $C_6\times D_6$ x 16, $C_6\times A_4$ x 16, $S_3\times A_4$ x 16, $C_3:S_4$ x 12, $C_2\times C_3^2:C_4$ x 11, $\SOPlus(4,2)$ x 8

Order 64: $C_2^3:D_4$ x 30, $C_2^4:C_4$ x 16, $C_2^3:D_4$ x 10, $C_2^4:C_4$ x 6, $C_2^4:C_4$ x 4, $D_4\times C_2^3$ x 4, $C_2^2.C_4^2$ x 4, $C_2^3.D_4$ x 2, $C_2^3:D_4$ x 2, $C_2^6$

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

Order 48: $C_2^2\times D_6$ x 296, $S_3\times D_4$ x 192, $C_6:D_4$ x 96, $C_2\times D_{12}$ x 49, $C_6\times D_4$ x 48, $C_4\times D_6$ x 48, $D_6:C_4$ x 48, $C_2\times S_4$ x 26, $C_2^2:C_{12}$ x 24, $C_6.D_4$ x 24, $C_2^3\times C_6$ x 20, $C_2^2\times C_{12}$ x 6, $C_6.C_2^3$ x 6, $C_2^2:A_4$ x 3, $C_2^2\times A_4$ x 3, $C_4\times A_4$ x 2

Order 36: $C_6:S_3$ x 67, $S_3^2$ x 28, $C_6\times S_3$ x 26, $C_6^2$ x 25, $C_3\times A_4$ x 16, $C_3:C_{12}$ x 12, $C_3^2:C_4$ x 4

Order 32: $C_2^2\wr C_2$ x 80, $C_2^2\times D_4$ x 74, $C_2^3:C_4$ x 61, $C_2^5$ x 17, $C_2^3:C_4$ x 16, $C_2^2.D_4$ x 16, $C_4:D_4$ x 16, $C_2^3\times C_4$ x 6, $C_2^2.D_4$ x 2

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 598, $C_3:D_4$ x 96, $C_2^2\times C_6$ x 90, $D_{12}$ x 52, $C_4\times S_3$ x 48, $C_3\times D_4$ x 48, $C_2\times C_{12}$ x 25, $C_6:C_4$ x 24, $S_4$ x 24, $C_2\times A_4$ x 17

Order 18: $C_3\times S_3$ x 16, $C_3:S_3$ x 12, $C_3\times C_6$ x 11

Order 16: $C_2\times D_4$ x 220, $C_2^4$ x 114, $C_2^2:C_4$ x 82, $C_2^2\times C_4$ x 46, $C_4:C_4$ x 8

Order 12: $D_6$ x 268, $C_2\times C_6$ x 95, $C_{12}$ x 14, $C_3:C_4$ x 12, $A_4$ x 11

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 241, $D_4$ x 104, $C_2\times C_4$ x 63

Order 6: $S_3$ x 32, $C_6$ x 27

Order 4: $C_2^2$ x 130, $C_4$ x 16

Order 3: $C_3$ x 5

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6912: $(C_2^3\times C_6^2):D_{12}$

Order 3456: $C_6^2:(C_2^2\times S_4)$, $(C_2^3\times C_6^2):C_{12}$, $(C_2^2\times C_6^2):D_{12}$

Order 2304: $C_2\times (C_2\times D_6:D_6).C_2^2$

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

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

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

Order 768: $C_2^5:D_{12}$

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

Order 432: $A_4:S_3^2$ x 4, $C_6:S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $C_6^2:A_4$, $C_2\times C_3^2:D_{12}$, $C_6^2:C_{12}$

Order 384: $C_2^5:C_{12}$, $C_2^4:D_{12}$, $C_2^2\wr S_3$, $C_2^5:D_6$

Order 288: $D_6^2:C_2$ x 24, $C_6^2:C_2^3$ x 10, $C_2\times C_6^2:C_4$ x 8, $C_2^3.S_3^2$ x 8, $C_6^2:D_4$ x 7, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 2, $C_2.D_6^2$ x 2, $D_6\wr C_2$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_2^5:C_3^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $S_3\times C_2^2:A_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_6^2:D_4$ x 2, $D_6\times S_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.D_4$, $C_2^3\times C_6^2$, $C_2\times D_6^2$, $(C_2\times C_6):S_4$

Order 256: $C_2^5:D_4$

Order 216: $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6:S_3^2$, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$

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

Order 144: $C_6^2:C_2^2$ x 48, $D_6:D_6$ x 32, $C_6^2:C_2^2$ x 12, $C_6:D_{12}$ x 12, $C_2^2.S_3^2$ x 12, $D_6^2$ x 10, $C_6^2:C_4$ x 8, $S_3^2:C_2^2$ x 6, $C_2^2\times C_6^2$ x 5, $C_6^2:C_4$ x 4, $C_6.D_{12}$ x 4, $C_6\times S_4$ x 4, $S_3\times S_4$ x 4, $C_6^2:C_4$ x 4, $A_4\times D_6$ x 3, $C_2^4:C_3^2$ x 2, $C_6:S_4$ x 2, $C_6^2:C_4$ x 2, $C_2.\SOPlus(4,2)$ x 2, $S_3^2:C_4$ x 2, $C_6^2:C_2^2$

Order 128: $C_2^4:D_4$ x 2, $C_2^3.C_4^2$, $C_2^4:D_4$, $C_2^4:D_4$, $C_2^5:C_4$, $C_2^5:C_4$

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

Order 96: $D_4\times D_6$ x 24, $C_2^3\times D_6$ x 15, $D_6.D_4$ x 8, $C_2^2:D_{12}$ x 7, $D_6:D_4$ x 7, $C_2^3:A_4$ x 4, $C_2^3:D_6$ x 4, $C_2^4:C_6$ x 3, $C_2^4:S_3$ x 3, $C_2^2.D_{12}$ x 3, $C_2^2:S_4$ x 2, $C_2^2\times S_4$ x 2, $C_{12}:C_2^3$ x 2, $C_2^2\times D_{12}$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3:C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^4\times C_6$, $C_2^3:C_{12}$, $C_4:S_4$

Order 72: $C_6:D_6$ x 85, $S_3\times D_6$ x 19, $C_2\times C_6^2$ x 15, $C_2\times C_3^2:C_4$ x 8, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6.D_6$ x 8, $C_6\times A_4$ x 6, $S_3\times A_4$ x 6, $C_6:C_{12}$ x 6, $C_6\times D_6$ x 5, $C_3\times S_4$ x 4, $C_3:S_4$ x 2, $\SOPlus(4,2)$ x 2

Order 64: $C_2^3:D_4$ x 10, $C_2^4:C_4$ x 4, $C_2^4:C_4$ x 3, $C_2^3:D_4$ x 3, $C_2^4:C_4$ x 2, $D_4\times C_2^3$ x 2, $C_2^3:D_4$ x 2, $C_2^6$, $C_2^2.C_4^2$, $C_2^3.D_4$

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

Order 48: $C_2^2\times D_6$ x 94, $S_3\times D_4$ x 32, $C_6:D_4$ x 24, $C_2\times D_{12}$ x 13, $C_6\times D_4$ x 12, $C_4\times D_6$ x 12, $C_2^3\times C_6$ x 8, $C_2\times S_4$ x 8, $D_6:C_4$ x 6, $C_2^2:C_{12}$ x 4, $C_6.D_4$ x 4, $C_2^2:A_4$ x 2, $C_2^2\times A_4$ x 2, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2, $C_4\times A_4$

Order 36: $C_6:S_3$ x 39, $C_6^2$ x 15, $C_6\times S_3$ x 8, $S_3^2$ x 8, $C_3\times A_4$ x 6, $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2

Order 32: $C_2^2\times D_4$ x 30, $C_2^3:C_4$ x 22, $C_2^5$ x 12, $C_2^2\wr C_2$ x 12, $C_2^3\times C_4$ x 4, $C_4:D_4$ x 4, $C_2^2.D_4$ x 2, $C_2^2.D_4$ x 2, $C_2^3:C_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 176, $C_2^2\times C_6$ x 30, $C_3:D_4$ x 16, $D_{12}$ x 9, $C_2\times A_4$ x 9, $C_4\times S_3$ x 8, $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 6, $S_4$ x 4

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

Order 16: $C_2^4$ x 64, $C_2\times D_4$ x 62, $C_2^2\times C_4$ x 22, $C_2^2:C_4$ x 14, $C_4:C_4$ x 2

Order 12: $D_6$ x 80, $C_2\times C_6$ x 31, $A_4$ x 5, $C_{12}$ x 3, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2^3$ x 123, $C_2\times C_4$ x 22, $D_4$ x 18

Order 6: $C_6$ x 12, $S_3$ x 11

Order 4: $C_2^2$ x 64, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 12

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6912: $(C_2^3\times C_6^2):D_{12}$ ($C_1$)

Order 3456: $(C_2^2\times C_6^2):D_{12}$ ($C_2$) x 4, $C_6^2:(C_2^2\times S_4)$ ($C_2$) x 2, $(C_2^3\times C_6^2):C_{12}$ ($C_2$)

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

Order 1152: $C_2\times C_6^2.(C_2^2\times C_4)$ ($S_3$)

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: $(C_2^2\times C_6^2):C_4$ ($D_6$) x 2, $C_6^2:C_2^4$ ($D_6$)

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

Order 288: $C_2^3\times C_6^2$ ($D_{12}$), $C_6^2:C_2^3$ ($D_{12}$), $C_6^2:C_2^3$ ($C_2\times D_6$), $C_2\times C_6^2:C_4$ ($S_4$)

Order 144: $C_6^2:C_4$ ($C_2\times S_4$) x 2, $C_2^2\times C_6^2$ ($C_2\times D_{12}$), $C_6^2:C_2^2$ ($C_2\times S_4$)

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

Order 72: $C_2\times C_6^2$ ($C_4:S_4$), $C_6:D_6$ ($C_2^2\times S_4$), $C_6:D_6$ ($C_4:S_4$)

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

Order 36: $C_6^2$ ($C_2^3:D_{12}$), $C_6:S_3$ ($C_2^3:S_4$)

Order 32: $C_2^5$ ($C_3^2:D_{12}$)

Order 18: $C_3\times C_6$ ($C_2^4:D_{12}$), $C_3:S_3$ ($C_2^4:D_{12}$), $C_3:S_3$ ($C_2^4:S_4$)

Order 16: $C_2^4$ ($C_2\times C_3^2:D_{12}$)

Order 9: $C_3^2$ ($C_2^5:D_{12}$)

Order 8: $C_2^3$ ($C_6^2:D_{12}$)

Order 4: $C_2^2$ ($C_2\times C_6^2:D_{12}$)

Order 2: $C_2$ ($(C_2^2\times C_6^2):D_{12}$)

Order 1: $C_1$ ($(C_2^3\times C_6^2):D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6912: $(C_2^3\times C_6^2):D_{12}$ ($C_1$)

Order 3456: $C_6^2:(C_2^2\times S_4)$ ($C_2$), $(C_2^3\times C_6^2):C_{12}$ ($C_2$), $(C_2^2\times C_6^2):D_{12}$ ($C_2$)

Order 1728: $C_6^2:(C_2^2\times A_4)$ ($C_2^2$), $C_6^2:(C_2\times S_4)$ ($C_2^2$), $(C_2^2\times C_6^2):C_{12}$ ($C_2^2$)

Order 1152: $C_2\times C_6^2.(C_2^2\times C_4)$ ($S_3$)

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: $C_6^2:C_2^4$ ($D_6$), $(C_2^2\times C_6^2):C_4$ ($D_6$)

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

Order 288: $C_2^3\times C_6^2$ ($D_{12}$), $C_6^2:C_2^3$ ($D_{12}$), $C_6^2:C_2^3$ ($C_2\times D_6$), $C_2\times C_6^2:C_4$ ($S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_2\times D_{12}$), $C_6^2:C_2^2$ ($C_2\times S_4$), $C_6^2:C_4$ ($C_2\times S_4$)

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

Order 72: $C_2\times C_6^2$ ($C_4:S_4$), $C_6:D_6$ ($C_2^2\times S_4$), $C_6:D_6$ ($C_4:S_4$)

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

Order 36: $C_6^2$ ($C_2^3:D_{12}$), $C_6:S_3$ ($C_2^3:S_4$)

Order 32: $C_2^5$ ($C_3^2:D_{12}$)

Order 18: $C_3\times C_6$ ($C_2^4:D_{12}$), $C_3:S_3$ ($C_2^4:D_{12}$), $C_3:S_3$ ($C_2^4:S_4$)

Order 16: $C_2^4$ ($C_2\times C_3^2:D_{12}$)

Order 9: $C_3^2$ ($C_2^5:D_{12}$)

Order 8: $C_2^3$ ($C_6^2:D_{12}$)

Order 4: $C_2^2$ ($C_2\times C_6^2:D_{12}$)

Order 2: $C_2$ ($(C_2^2\times C_6^2):D_{12}$)

Order 1: $C_1$ ($(C_2^3\times C_6^2):D_{12}$)

Series

Derived series

$(C_2^3\times C_6^2):D_{12}$

$\rhd$

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

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$(C_2^3\times C_6^2):D_{12}$

$\rhd$

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

$\rhd$

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

$\rhd$

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

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$(C_2^3\times C_6^2):D_{12}$

$\rhd$

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

$\rhd$

$C_6^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 9 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 $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $82 \times 82$ rational character table.