Abstract group 6912.hz: $C_6^2:D_4:D_{12}$

• Label: 6912.hz
• Order: \( 2^{8} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $14$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_6^2:D_4:D_{12}$
Order:	\(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^2.(C_2\times A_4).D_4^2$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
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	487	296	1560	1592	2976	6912
Conjugacy classes	1	13	5	16	17	14	66
Divisions	1	13	5	16	17	12	64
Autjugacy classes	1	8	3	9	8	5	34

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	8	10	8	8	10	4	10	8	66
Irr. rational chars.	8	6	8	10	10	4	10	8	64

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$1612800$

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:	${\langle a, b, c, d, e, f \mid b^{12}=c^{6}=d^{12}=e^{2}=f^{2}=[a,e]=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(5,13,8,11,12,9)(10,14), (1,3)(2,4)(5,10)(6,7)(9,11,13,14), (9,13)(11,14) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2:D_4)$ $\,\rtimes\,$ $D_{12}$	$(C_6^2:D_4)$ $\,\rtimes\,$ $D_{12}$	$C_2^3$ $\,\rtimes\,$ $(C_6^2:D_{12})$	$(C_6:D_6)$ $\,\rtimes\,$ $(C_4:S_4)$	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2.C_2^4)$ . $D_6$	$C_6^2$ . $(C_2^3:D_{12})$	$(C_3:S_3)$ . $(C_2^4:S_4)$	$(C_6:D_6)$ . $(C_2^2\times S_4)$	all 10

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 97456 subgroups in 2726 conjugacy classes, 45 normal (35 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_6^2:D_4:D_{12}$
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:	$\operatorname{Fit} \simeq$ $C_6^2:D_4$	$G/\operatorname{Fit} \simeq$ $D_{12}$
Radical:	$R \simeq$ $C_6^2:D_4:D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^3:D_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2.D_4^2$
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

Order 6912: $C_6^2:D_4:D_{12}$

Order 3456: $(C_2^2\times C_6^2):D_{12}$ x 2, $C_6^2.(C_2^2\times S_4)$ x 2, $(C_2^2\times C_6^2):D_{12}$ x 2, $C_6^2:D_4:C_{12}$

Order 2304: $C_6^2.D_4^2$

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

Order 1152: $D_6^2:D_4$ x 2, $D_6^2.C_2^3$ 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, $D_6^2.D_4$ x 2, $S_3\times C_2^3:S_4$ x 2, $C_6^2:(C_4\times D_4)$, $C_6:D_{12}.Q_8$, $D_6^2.C_2^3$

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

Order 768: $(C_2^3\times C_4):S_4$

Order 576: $D_6^2.C_2^2$ x 8, $D_6^2.C_2^2$ x 5, $C_2^4:S_3^2$ x 4, $D_6^2.C_2^2$ x 4, $C_2^4:S_3^2$ x 4, $C_3\times C_2^3:S_4$ x 4, $C_2^3.\SOPlus(4,2)$ x 4, $D_6^2:C_4$ x 4, $C_2^4:S_3^2$ x 4, $D_6^2:C_4$ x 4, $D_6^2:C_2^2$ x 3, $D_6^2:C_2^2$ x 2, $D_4:\SOPlus(4,2)$ x 2, $C_3^2:D_4^2$ x 2, $S_3\times C_2^3:A_4$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_3\times C_2^4:D_6$ 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, $C_2^3.\SOPlus(4,2)$ x 2, $C_2^2\times C_6^2:C_4$, $C_6^2:(C_2^2\times C_4)$, $(C_6\times C_{12}):D_4$, $(C_6\times C_{12}):D_4$, $C_6^2.C_2^4$, $(C_2^2\times C_6^2):C_4$, $C_6^2.(C_2^2\times C_4)$, $C_6^2.(C_2^2\times C_4)$, $C_6^2:C_4^2$

Order 432: $A_4:S_3^2$ x 10, $C_6^2:C_{12}$ x 4, $C_6^2:D_6$ x 2, $C_6^2:A_4$, $C_6:S_3\times A_4$, $C_2\times C_3^2:D_{12}$

Order 384: $C_2^4:D_{12}$ x 2, $C_2^4:D_{12}$ x 2, $C_2^4:S_4$ x 2, $C_2\wr C_2^2\times S_3$ x 2, $C_4\times C_2^3:A_4$

Order 288: $C_6^2:D_4$ x 24, $D_6^2:C_2$ x 10, $C_6^2.D_4$ x 10, $D_6\wr C_2$ x 8, $C_6^2.D_4$ x 8, $C_6^2:D_4$ x 8, $C_2^3.S_3^2$ x 8, $C_3\times C_2^2:S_4$ x 8, $C_2\times C_6^2:C_4$ x 7, $C_6^2.D_4$ x 6, $C_6^2.D_4$ x 5, $C_2^3:S_3^2$ x 4, $D_{12}:D_6$ x 4, $D_4:S_3^2$ x 4, $D_4\times S_3^2$ x 4, $C_3^2:C_4\times D_4$ x 4, $S_3^2:D_4$ x 4, $C_4\times \SOPlus(4,2)$ x 4, $S_3^2:Q_8$ x 4, $C_6^2:D_4$ x 4, $C_3\times C_2^3:A_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2.D_4$ x 4, $(C_2\times C_6):S_4$ x 4, $C_2\times C_6^2:C_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 2, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $C_4.\SOPlus(4,2)$ x 2, $(C_2\times D_6):A_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $S_3\times C_2^2:A_4$ x 2, $D_6\times S_4$ x 2, $(C_6\times C_{12}):C_4$, $C_2\times C_3^2:C_4^2$, $C_4:\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $(C_6\times C_{12}):C_4$, $C_6^2:C_2^3$, $C_6^2:C_2^3$

Order 256: $C_2^2.D_4^2$

Order 216: $C_3^2:S_4$ x 8, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_6:S_3^2$ x 2, $C_6^2:C_6$, $C_2\times C_3^2:C_{12}$

Order 192: $C_2^3:S_4$ x 6, $C_2^4:D_6$ x 6, $C_2^3:C_4\times S_3$ x 6, $C_2^3:D_{12}$ x 6, $C_2^4:D_6$ x 6, $C_2^3:S_4$ x 4, $C_3\times C_2\wr C_2^2$ x 2, $(C_2\times C_{12}):D_4$ x 2, $D_{12}:C_2^3$ x 2, $C_2^4:C_{12}$, $C_2^4:C_{12}$, $C_2^3:D_{12}$, $C_2^4:A_4$

Order 144: $D_6:D_6$ x 40, $C_6^2:C_4$ x 26, $S_3^2:C_4$ x 24, $S_3^2:C_2^2$ x 14, $S_3\times S_4$ x 12, $C_2.\SOPlus(4,2)$ x 12, $C_6^2:C_2^2$ x 10, $C_6:D_{12}$ x 10, $C_2^2.S_3^2$ x 10, $C_6^2:C_4$ x 8, $C_6.D_{12}$ x 8, $D_6:D_6$ x 8, $D_6.D_6$ x 8, $C_6^2:C_2^2$ x 7, $D_6^2$ x 6, $C_6^2:C_4$ x 6, $C_2^4:C_3^2$ x 4, $C_6\times S_4$ x 4, $C_{12}:D_6$ x 4, $C_{12}.D_6$ x 4, $C_{12}:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_4\times S_3^2$ x 4, $C_{12}.D_6$ x 4, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6^2:C_2^2$ x 2, $D_{12}:C_6$ x 2, $D_6.D_6$ x 2, $C_{12}:D_6$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $D_{12}:S_3$ x 2, $(C_3\times C_{12}):C_4$ x 2, $C_3^2:C_4^2$ x 2, $C_6.D_{12}$, $C_2^2\times C_6^2$, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$, $C_6^2:C_4$, $C_6^2:C_4$

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

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 16, $D_4:D_6$ x 12, $C_2^2.D_{12}$ x 12, $C_2^2:D_{12}$ x 10, $D_6.D_4$ x 10, $D_6:D_4$ x 10, $C_2^2:S_4$ x 8, $C_2^3:C_{12}$ x 6, $C_2^3.D_6$ x 6, $D_4:D_6$ x 6, $D_4:D_6$ x 6, $C_4:S_4$ x 6, $C_2^4:C_6$ x 6, $C_2^4:S_3$ x 6, $C_2^3:A_4$ x 4, $C_2^3\times D_6$ x 2, $C_2^2\times S_4$ x 2, $C_6.C_2^4$ x 2, $C_2^3:A_4$, $C_2^3:C_{12}$

Order 72: $C_6:D_6$ x 25, $S_3\times D_6$ x 24, $C_2\times C_3^2:C_4$ x 22, $C_6\wr C_2$ x 20, $C_3:D_{12}$ x 20, $C_6.D_6$ x 18, $C_3\times S_4$ x 16, $\SOPlus(4,2)$ x 10, $C_6:C_{12}$ x 10, $S_3\times A_4$ x 8, $C_3:S_4$ x 8, $C_6\times D_6$ x 6, $C_2\times C_6^2$ x 4, $C_6\times A_4$ x 4, $C_6^2:C_2$ x 4, $S_3\times C_{12}$ x 4, $C_6.D_6$ x 4, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_3\times D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3^2:Q_8$ x 2, $D_6:S_3$ x 2, $C_6\times C_{12}$, $C_6.D_6$

Order 64: $C_2^4:C_4$ x 10, $C_2^3.D_4$ x 9, $C_2\wr C_2^2$ x 8, $C_2^3:D_4$ x 5, $C_2^3:D_4$ x 5, $C_2^3.D_4$ x 4, $D_4:D_4$ x 4, $C_2^3:D_4$ x 3, $D_4:C_2^3$ x 2, $C_2^2.C_4^2$ x 2, $D_4:D_4$ x 2, $D_4^2$ x 2, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2:C_4^2$, $C_2^4\times C_4$, $C_2^2.C_2^4$, $C_4^2.C_2^2$, $C_4^2:C_2^2$

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

Order 48: $S_3\times D_4$ x 70, $C_6:D_4$ x 32, $C_2^2\times D_6$ x 20, $D_4:S_3$ x 18, $D_{12}:C_2$ x 18, $C_2\times D_{12}$ x 17, $C_6\times D_4$ x 16, $C_4\times D_6$ x 16, $C_2\times S_4$ x 14, $D_6:C_4$ x 14, $C_2^2:C_{12}$ x 10, $C_6.D_4$ x 10, $D_4:C_6$ x 6, $D_{12}:C_2$ x 6, $C_4\times A_4$ x 4, $C_2^2:A_4$ x 3, $C_2^3\times C_6$ x 2, $S_3\times Q_8$ x 2, $C_2^2\times A_4$

Order 36: $C_6:S_3$ x 22, $S_3^2$ x 22, $C_6\times S_3$ x 16, $C_3\times A_4$ x 12, $C_6^2$ x 9, $C_3:C_{12}$ x 8, $C_3^2:C_4$ x 6, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_4:D_4$ x 28, $C_2^2\wr C_2$ x 24, $C_2^3:C_4$ x 22, $C_2^2\times D_4$ x 18, $C_2^3:C_4$ x 16, $D_4:C_2^2$ x 12, $C_2^2.D_4$ x 8, $C_4\times D_4$ x 8, $C_2^3\times C_4$ x 7, $C_2.C_4^2$ x 7, $D_4:C_2^2$ x 6, $C_2^2.D_4$ x 6, $C_2^2:Q_8$ x 4, $C_4^2:C_2$ x 2, $C_2^5$, $C_4:Q_8$, $C_4:D_4$, $C_2\times C_4^2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 82, $C_3:D_4$ x 52, $C_4\times S_3$ x 32, $D_{12}$ x 30, $C_3\times D_4$ x 26, $C_2\times C_{12}$ x 17, $C_6:C_4$ x 16, $S_4$ x 16, $C_2^2\times C_6$ x 14, $C_2\times A_4$ x 7, $C_3:Q_8$ x 6, $C_3\times Q_8$ x 2

Order 18: $C_3\times S_3$ x 16, $C_3:S_3$ x 9, $C_3\times C_6$ x 8

Order 16: $C_2\times D_4$ x 100, $C_2^2:C_4$ x 49, $C_2^2\times C_4$ x 41, $D_4:C_2$ x 24, $C_4:C_4$ x 14, $C_2^4$ x 14, $C_4^2$ x 2, $C_2\times Q_8$ x 2

Order 12: $D_6$ x 76, $C_2\times C_6$ x 29, $C_{12}$ x 12, $C_3:C_4$ x 10, $A_4$ x 9

Order 9: $C_3^2$ x 5

Order 8: $D_4$ x 62, $C_2\times C_4$ x 62, $C_2^3$ x 57, $Q_8$ x 4

Order 6: $S_3$ x 22, $C_6$ x 17

Order 4: $C_2^2$ x 53, $C_4$ x 16

Order 3: $C_3$ x 5

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 6912: $C_6^2:D_4:D_{12}$

Order 3456: $(C_2^2\times C_6^2):D_{12}$, $C_6^2.(C_2^2\times S_4)$, $C_6^2:D_4:C_{12}$, $(C_2^2\times C_6^2):D_{12}$

Order 2304: $C_6^2.D_4^2$

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

Order 1152: $D_6^2:D_4$, $D_6^2.C_2^3$, $C_6^2:(C_4\times D_4)$, $D_6^2.D_4$, $D_6^2:D_4$, $C_6^2.(C_4\times D_4)$, $C_6:D_{12}.Q_8$, $D_6^2.D_4$, $S_3\times C_2^3:S_4$, $D_6^2.C_2^3$

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

Order 768: $(C_2^3\times C_4):S_4$

Order 576: $D_6^2.C_2^2$ x 3, $D_6^2:C_2^2$ x 2, $C_3\times C_2^3:S_4$ x 2, $C_2^3.\SOPlus(4,2)$ x 2, $D_6^2.C_2^2$ x 2, $C_2^2\times C_6^2:C_4$, $C_2^4:S_3^2$, $D_6^2:C_2^2$, $C_6^2:(C_2^2\times C_4)$, $D_4:\SOPlus(4,2)$, $D_6^2.C_2^2$, $C_3^2:D_4^2$, $(C_6\times C_{12}):D_4$, $(C_6\times C_{12}):D_4$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $(C_2^2\times C_6^2):C_4$, $C_6^2.(C_2^2\times C_4)$, $C_6^2.(C_2^2\times C_4)$, $C_6^2:C_4^2$, $S_3\times C_2^3:A_4$, $(C_2^2\times C_6):S_4$, $D_6^2:C_4$, $C_2^4:S_3^2$, $C_3\times C_2^4:D_6$, $C_2^4:S_3^2$, $(C_2\times C_{12}):D_{12}$, $(C_6\times D_4).D_6$, $(C_2\times D_6):D_{12}$, $D_6^2:C_4$, $C_2^3.\SOPlus(4,2)$

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

Order 384: $C_4\times C_2^3:A_4$, $C_2^4:D_{12}$, $C_2^4:D_{12}$, $C_2^4:S_4$, $C_2\wr C_2^2\times S_3$

Order 288: $C_6^2:D_4$ x 12, $C_2\times C_6^2:C_4$ x 5, $D_6^2:C_2$ x 3, $C_2\times C_6^2:C_4$ x 3, $C_6^2.D_4$ x 3, $C_6^2.D_4$ x 3, $C_2^3:S_3^2$ x 2, $C_3^2:C_4\times D_4$ x 2, $D_6\wr C_2$ x 2, $C_6^2.D_4$ x 2, $C_4\times \SOPlus(4,2)$ x 2, $C_3\times C_2^3:A_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2.D_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$, $D_{12}:D_6$, $D_4:S_3^2$, $D_4\times S_3^2$, $D_{12}:D_6$, $D_{12}:D_6$, $(C_6\times C_{12}):C_4$, $C_2\times C_3^2:C_4^2$, $C_4:\SOPlus(4,2)$, $S_3^2:D_4$, $C_4.\SOPlus(4,2)$, $C_4.\SOPlus(4,2)$, $S_3^2:Q_8$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2:D_4$, $C_6^2.C_2^3$, $C_6^2:D_4$, $(C_6\times C_{12}):C_4$, $(C_2\times D_6):A_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2:C_2^3$, $S_3\times C_2^2:A_4$, $(C_2\times C_6):S_4$, $D_6\times S_4$, $C_6^2:C_2^3$

Order 256: $C_2^2.D_4^2$

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

Order 192: $C_2^3:S_4$ x 3, $C_2^4:D_6$ x 2, $C_2^3:C_4\times S_3$ x 2, $C_2^3:D_{12}$ x 2, $C_2^4:D_6$ x 2, $C_3\times C_2\wr C_2^2$, $(C_2\times C_{12}):D_4$, $C_2^4:C_{12}$, $C_2^3:S_4$, $D_{12}:C_2^3$, $C_2^4:C_{12}$, $C_2^3:D_{12}$, $C_2^4:A_4$

Order 144: $C_6^2:C_4$ x 17, $D_6:D_6$ x 11, $S_3^2:C_2^2$ x 8, $C_2.\SOPlus(4,2)$ x 6, $S_3^2:C_4$ x 6, $C_6^2:C_2^2$ x 5, $S_3\times S_4$ x 3, $C_6^2:C_2^2$ x 3, $C_6:D_{12}$ x 3, $C_2^2.S_3^2$ x 3, $C_6^2:C_4$ x 3, $C_6^2:C_4$ x 2, $C_6.D_{12}$ x 2, $C_2^4:C_3^2$ x 2, $D_6^2$ x 2, $C_6\times S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_3^2:C_4^2$ x 2, $C_6.D_{12}$, $C_2^2\times C_6^2$, $A_4\times D_6$, $C_6:S_4$, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$, $C_{12}.D_6$, $C_{12}:D_6$, $D_{12}:C_6$, $D_6.D_6$, $C_{12}:D_6$, $S_3\times D_{12}$, $C_4\times S_3^2$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $D_{12}:S_3$, $(C_3\times C_{12}):C_4$, $C_6^2:C_4$, $C_6^2:C_4$

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

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

Order 96: $D_4\times D_6$ x 5, $D_4:D_6$ x 4, $C_2^2:D_{12}$ x 3, $D_6.D_4$ x 3, $C_2^3:A_4$ x 3, $C_4:S_4$ x 3, $D_6:D_4$ x 3, $C_2^2.D_{12}$ x 3, $C_2^3:C_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^2:S_4$ x 2, $D_4:D_6$ x 2, $D_4:D_6$ x 2, $C_2^4:C_6$ x 2, $C_2^4:S_3$ x 2, $C_2^3\times D_6$, $C_2^3:A_4$, $C_2^2\times S_4$, $C_6.C_2^4$, $C_2^3:C_{12}$

Order 72: $C_6:D_6$ x 16, $C_2\times C_3^2:C_4$ x 15, $S_3\times D_6$ x 8, $\SOPlus(4,2)$ x 5, $C_6\wr C_2$ x 5, $C_3:D_{12}$ x 5, $C_6.D_6$ x 5, $C_3\times S_4$ x 4, $C_2\times C_6^2$ x 3, $S_3\times A_4$ x 3, $C_6:C_{12}$ x 3, $C_6\times D_6$ x 2, $C_6\times A_4$ x 2, $C_3:S_4$ x 2, $C_6^2:C_2$ x 2, $C_{12}:S_3$ x 2, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_6.D_6$, $C_3:D_{12}$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$

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

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

Order 48: $S_3\times D_4$ x 18, $C_6:D_4$ x 10, $C_2^2\times D_6$ x 7, $C_2\times D_{12}$ x 6, $C_2\times S_4$ x 5, $C_6\times D_4$ x 5, $D_4:S_3$ x 5, $D_{12}:C_2$ x 5, $C_4\times D_6$ x 5, $D_6:C_4$ x 4, $C_4\times A_4$ x 3, $C_2^2:C_{12}$ x 3, $C_6.D_4$ x 3, $C_2^2:A_4$ x 2, $D_4:C_6$ x 2, $D_{12}:C_2$ x 2, $C_2^3\times C_6$, $C_2^2\times A_4$, $S_3\times Q_8$

Order 36: $C_6:S_3$ x 14, $C_6^2$ x 6, $C_6\times S_3$ x 6, $S_3^2$ x 6, $C_3^2:C_4$ x 5, $C_3\times A_4$ x 4, $C_3:C_{12}$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_4:D_4$ x 13, $C_2^2\wr C_2$ x 9, $C_2^3:C_4$ x 9, $C_2^2\times D_4$ x 7, $C_2^3:C_4$ x 5, $D_4:C_2^2$ x 5, $C_2^3\times C_4$ x 5, $C_4\times D_4$ x 4, $C_2^2.D_4$ x 4, $C_2.C_4^2$ x 3, $D_4:C_2^2$ x 2, $C_2^2.D_4$ x 2, $C_2^5$, $C_4:Q_8$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_2\times C_4^2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 26, $C_3:D_4$ x 13, $D_{12}$ x 9, $C_4\times S_3$ x 9, $C_3\times D_4$ x 7, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 5, $C_2^2\times C_6$ x 5, $C_2\times A_4$ x 5, $S_4$ x 4, $C_3:Q_8$ x 2, $C_3\times Q_8$

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

Order 16: $C_2\times D_4$ x 36, $C_2^2\times C_4$ x 22, $C_2^2:C_4$ x 16, $C_2^4$ x 8, $D_4:C_2$ x 8, $C_4:C_4$ x 7, $C_4^2$ x 2, $C_2\times Q_8$

Order 12: $D_6$ x 24, $C_2\times C_6$ x 10, $C_{12}$ x 5, $A_4$ x 4, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 29, $C_2^3$ x 28, $D_4$ x 21, $Q_8$ x 2

Order 6: $C_6$ x 8, $S_3$ x 7

Order 4: $C_2^2$ x 26, $C_4$ x 9

Order 3: $C_3$ x 3

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6912: $C_6^2:D_4:D_{12}$ ($C_1$)

Order 3456: $(C_2^2\times C_6^2):D_{12}$ ($C_2$) x 2, $C_6^2.(C_2^2\times S_4)$ ($C_2$) x 2, $(C_2^2\times C_6^2):D_{12}$ ($C_2$) x 2, $C_6^2:D_4:C_{12}$ ($C_2$)

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

Order 1152: $C_6^2:(C_4\times D_4)$ ($S_3$)

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

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

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

Order 288: $C_6^2:D_4$ ($D_{12}$), $C_6^2:D_4$ ($D_{12}$), $C_6^2:C_2^3$ ($C_2\times D_6$), $C_2\times C_6^2:C_4$ ($S_4$)

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: $C_3^2:C_4$ ($C_2^3:S_4$), $C_6^2$ ($C_2^3:D_{12}$)

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

Order 18: $C_3:S_3$ ($C_2^4:S_4$)

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

Order 9: $C_3^2$ ($(C_2^3\times C_4):S_4$)

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

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

Order 1: $C_1$ ($C_6^2:D_4:D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6912: $C_6^2:D_4:D_{12}$ ($C_1$)

Order 3456: $(C_2^2\times C_6^2):D_{12}$ ($C_2$), $C_6^2.(C_2^2\times S_4)$ ($C_2$), $C_6^2:D_4:C_{12}$ ($C_2$), $(C_2^2\times C_6^2):D_{12}$ ($C_2$)

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

Order 1152: $C_6^2:(C_4\times D_4)$ ($S_3$)

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

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

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

Order 288: $C_6^2:D_4$ ($D_{12}$), $C_6^2:D_4$ ($D_{12}$), $C_6^2:C_2^3$ ($C_2\times D_6$), $C_2\times C_6^2:C_4$ ($S_4$)

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: $C_3^2:C_4$ ($C_2^3:S_4$), $C_6^2$ ($C_2^3:D_{12}$)

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

Order 18: $C_3:S_3$ ($C_2^4:S_4$)

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

Order 9: $C_3^2$ ($(C_2^3\times C_4):S_4$)

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

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

Order 1: $C_1$ ($C_6^2:D_4:D_{12}$)

Series

Derived series

$C_6^2:D_4:D_{12}$

$\rhd$

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

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^2:D_4:D_{12}$

$\rhd$

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

$\rhd$

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

$\rhd$

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

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_6^2:D_4:D_{12}$

$\rhd$

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

$\rhd$

$C_6^2:A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $66 \times 66$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $64 \times 64$ rational character table.