Abstract group 6912.ln: $C_3:S_4\times \GL(2,\mathbb{Z}/4)$

• Label: 6912.ln
• Order: \( 2^{8} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$C_3:S_4\times \GL(2,\mathbb{Z}/4)$
Order:	\(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_3.(C_3\times A_4^2).C_2^5$, of order \(82944\)\(\medspace = 2^{10} \cdot 3^{4} \)
Composition factors:	$C_2$ x 8, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	615	242	1944	2382	1728	6912
Conjugacy classes	1	20	9	19	58	19	126
Divisions	1	20	9	19	50	18	117
Autjugacy classes	1	18	5	17	33	11	85

Dimension	1	2	3	4	6	8	9	12	18	24	36
Irr. complex chars.	8	26	16	20	32	0	8	9	6	0	1	126
Irr. rational chars.	8	22	16	14	28	4	8	9	6	1	1	117

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{12}=d^{2}=e^{6}=f^{2}=g^{2}=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(4,7,5,6)(10,13,11), (1,2,3)(4,6,7)(8,12)(14,15), (1,2)(5,7), (4,5,7) \!\cdots\! \rangle$
Direct product:	$(C_3:S_4)$ $\, \times\, $ $\GL(2,\mathbb{Z}/4)$
Semidirect product:	$(A_4^2:S_3)$ $\,\rtimes\,$ $D_4$	$A_4^2$ $\,\rtimes\,$ $(S_3\times D_4)$	$(C_6.S_4^2)$ $\,\rtimes\,$ $C_2$	$(C_6.S_4^2)$ $\,\rtimes\,$ $C_2$	all 64
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_2\times S_4^2)$	$C_2^5$ . $(C_6:S_3^2)$	$(A_4^2:D_6)$ . $C_2^2$	$(C_6\times A_4^2)$ . $C_2^3$	all 18

Elements of the group are displayed as permutations of degree 15.

Homology

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

Subgroups

There are 231128 subgroups in 7642 conjugacy classes, 119 normal (77 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times A_4^2.D_6$
Commutator:	$G' \simeq$ $C_6\times A_4^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times A_4^2.D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5\times C_6$	$G/\operatorname{Fit} \simeq$ $S_3^2$
Radical:	$R \simeq$ $C_3:S_4\times \GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times D_6$
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

All subgroups of index up to 64 (order at least 108) are shown, as well as all normal subgroups of any index.

Order 6912: $C_3:S_4\times \GL(2,\mathbb{Z}/4)$

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

Order 2304: $S_4\times \GL(2,\mathbb{Z}/4)$ x 3, $(A_4\times C_2^4):D_6$, $C_2^6:S_3^2$

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

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

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

Order 768: $C_2^6:D_6$ x 3, $C_2^6:D_6$, $D_4\times \GL(2,\mathbb{Z}/4)$

Order 576: $A_4^2:C_2^2$ x 15, $D_4\times C_3:S_4$ x 12, $C_2^2\times C_6:S_4$ x 11, $C_6:\GL(2,\mathbb{Z}/4)$ x 9, $C_6:D_4\times A_4$ x 8, $C_2^4:S_3^2$ x 7, $S_3\times \GL(2,\mathbb{Z}/4)$ x 7, $C_2^2\times A_4^2$ x 6, $A_4^2:C_2^2$ x 6, $S_4^2$ x 6, $A_4^2:C_4$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_6\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_6^2$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_6:\GL(2,\mathbb{Z}/4)$ x 3, $A_4^2:C_4$ x 3, $C_2^4.C_6^2$ x 3, $C_2^4:C_6^2$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:C_6^2$, $C_2^2\times A_4\times D_6$, $(C_2^2\times C_6):S_4$, $C_2^2\times C_6\times S_4$, $C_2^2:D_6^2$, $C_6.C_2^3\times A_4$, $(C_2^2\times C_6).S_4$, $C_2^2\times A_4:C_{12}$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$

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

Order 384: $C_2^2:\GL(2,\mathbb{Z}/4)$ x 11, $\GL(2,\mathbb{Z}/4):C_2^2$ x 10, $C_2^4:D_{12}$ x 9, $C_2^5.D_6$ x 9, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4\times S_4$ x 3, $C_2\wr S_3$ x 3, $C_6:D_4^2$ x 3, $C_2^6:C_6$ x 3, $C_2^5:D_6$ x 3, $C_6.D_4^2$ x 3, $C_2^5:D_6$ x 3, $C_6.D_4^2$ x 3, $C_6.D_4^2$ x 3, $C_2^4:D_{12}$ x 3, $C_6.D_4^2$ x 3, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:C_2^5$, $C_2^6:C_6$, $C_2^6:C_6$, $C_2^5:D_6$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_2^5.D_6$, $C_4:\GL(2,\mathbb{Z}/4)$, $C_4\times \GL(2,\mathbb{Z}/4)$, $C_2^6:S_3$, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^5.D_6$

Order 288: $C_2\times C_6:S_4$ x 41, $C_3:\GL(2,\mathbb{Z}/4)$ x 32, $A_4\times C_3:D_4$ x 13, $C_2\times A_4^2$ x 12, $A_4\times S_4$ x 12, $D_6:S_4$ x 11, $C_2^3:C_6^2$ x 11, $D_6\times S_4$ x 11, $C_2\times A_4\times D_6$ x 10, $C_3\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2\times C_6\times S_4$ x 7, $C_2^3.C_6^2$ x 6, $C_2\times C_6.S_4$ x 6, $C_{12}:S_4$ x 6, $C_{12}:S_4$ x 6, $\PSOPlus(4,3)$ x 6, $D_6:S_4$ x 5, $D_6.S_4$ x 5, $C_2^3.S_3^2$ x 5, $C_2^3.S_3^2$ x 5, $C_6:C_4\times A_4$ x 4, $C_2^3.C_6^2$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $C_6^2:C_2^3$ x 3, $D_6^2:C_2$ x 2, $C_2\times A_4:C_{12}$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^5:C_3^2$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $D_6^2:C_2$, $C_2^3.S_3^2$, $C_6^2:C_2^3$, $(C_2\times C_6):S_4$

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

Order 216: $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 3, $C_3^2:S_4$ x 3, $C_3^2:S_4$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$, $C_3^2\times S_4$, $C_6^2:S_3$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_6.S_3^2$

Order 192: $D_4\times S_4$ x 40, $C_2\times \GL(2,\mathbb{Z}/4)$ x 36, $C_2^3\times S_4$ x 35, $C_2^4:D_6$ x 25, $C_2.\GL(2,\mathbb{Z}/4)$ x 21, $D_6:C_2^4$ x 21, $C_2.\GL(2,\mathbb{Z}/4)$ x 17, $C_2^5:C_6$ x 17, $C_2^4.D_6$ x 12, $C_2^4:D_6$ x 12, $C_2^4:D_6$ x 11, $C_2^3:D_{12}$ x 10, $C_2^4.D_6$ x 10, $C_2^4:D_6$ x 10, $C_2^4:C_{12}$ x 9, $C_2^5:C_6$ x 8, $C_2^4.D_6$ x 7, $C_2^4.D_6$ x 6, $C_2^3.D_{12}$ x 6, $C_2^3.D_{12}$ x 6, $C_2^3:D_{12}$ x 6, $C_2^4.D_6$ x 6, $C_2^4.D_6$ x 6, $C_2^3.D_{12}$ x 6, $A_4\times C_2^4$ x 6, $C_2^4.D_6$ x 6, $C_2^3.D_{12}$ x 3, $C_2^3.D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^2\wr C_3$ x 3, $C_2^3:S_4$ x 3, $C_{12}:C_2^4$ x 3, $C_{12}:C_2^4$ x 3, $C_2^3.S_4$ x 3, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3.S_4$ x 2, $C_2^3.D_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $C_2^5\times C_6$, $D_6\times C_2^4$, $C_6.C_2^5$, $C_2^4:C_{12}$

Order 144: $C_6:S_4$ x 53, $C_2^2:C_6^2$ x 37, $A_4\times D_6$ x 26, $C_6\times S_4$ x 21, $C_6.S_4$ x 21, $S_3\times S_4$ x 16, $C_{12}:D_6$ x 12, $C_6^2:C_2^2$ x 12, $C_6^2:C_2^2$ x 11, $D_6:D_6$ x 11, $C_6^2:C_2^2$ x 9, $C_6.D_{12}$ x 6, $A_4^2$ x 6, $A_4\times C_3:C_4$ x 6, $C_6^2:C_4$ x 6, $A_4:C_{12}$ x 5, $C_6.D_{12}$ x 4, $C_4:C_6^2$ x 3, $C_6:D_{12}$ x 3, $C_{12}:D_6$ x 3, $C_{12}\times A_4$ x 3, $D_6:D_6$ x 3, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 2, $C_2^2\times C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$ x 2, $C_6:D_{12}$ x 2, $D_6.D_6$ x 2, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_2^4:C_3^2$, $D_6^2$

Order 128: $C_2^4:D_4$ x 6, $C_2^4:D_4$ x 6, $C_2\times D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2^4:D_4$ x 2, $C_2^3\wr C_2$ x 2, $D_4\times C_2^4$, $C_2^4:D_4$, $C_2^3.C_2^4$

Order 108: $C_3^2:D_6$ x 4, $C_3^2\times A_4$ x 2, $C_3:S_3^2$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 23, $C_2^3\times A_4$ x 4, $C_2^4\times C_6$

Order 72: $C_3:S_4$ x 12, $C_6\times A_4$ x 2

Order 64: $C_2^6$

Order 48: $S_3\times D_4$ x 56, $C_2\times S_4$ x 53, $C_2^2\times A_4$ x 8, $C_6:D_4$ x 4, $C_2^3\times C_6$ x 3, $A_4:C_4$

Order 36: $C_6:S_3$ x 16, $S_3^2$ x 10, $C_3\times A_4$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 24: $S_4$ x 34, $C_3:D_4$ x 20, $C_2\times A_4$ x 4, $C_2^2\times C_6$ x 2

Order 18: $C_3:S_3$ x 11

Order 16: $C_2\times D_4$ x 32, $C_2^4$ x 3

Order 12: $D_6$ x 84, $A_4$ x 4, $C_2\times C_6$ x 3

Order 8: $D_4$ x 52, $C_2^3$ x 2

Order 6: $S_3$ x 35, $C_6$

Order 4: $C_2^2$ x 35

Order 3: $C_3$

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 64 (order at least 108) are shown, as well as all normal subgroups of any index.

Order 6912: $C_3:S_4\times \GL(2,\mathbb{Z}/4)$

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

Order 2304: $(A_4\times C_2^4):D_6$, $C_2^6:S_3^2$, $S_4\times \GL(2,\mathbb{Z}/4)$

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

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

Order 864: $C_6\times A_4^2$ x 2, $A_4^2:S_3$ x 2, $C_6^2:S_4$ x 2, $C_6:D_6\times A_4$, $C_6^2:S_4$, $S_3\times C_6:S_4$, $C_6:S_3\times S_4$, $A_4^2:S_3$, $A_4^2:C_6$, $C_6\wr C_2\times A_4$, $C_3^2\times \GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_3^2:(C_4\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$

Order 768: $C_2^6:D_6$, $D_4\times \GL(2,\mathbb{Z}/4)$, $C_2^6:D_6$

Order 576: $C_2^2\times C_6:S_4$ x 9, $C_6:\GL(2,\mathbb{Z}/4)$ x 9, $D_4\times C_3:S_4$ x 7, $C_6:D_4\times A_4$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $S_3\times \GL(2,\mathbb{Z}/4)$ x 5, $A_4^2:C_2^2$ x 4, $C_6\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_6^2$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2^4:S_3^2$ x 3, $C_6:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.C_6^2$ x 3, $C_2^4:C_6^2$ x 2, $C_2^2\times A_4^2$ x 2, $A_4^2:C_2^2$ x 2, $A_4^2:C_4$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:C_6^2$, $C_2^2\times A_4\times D_6$, $(C_2^2\times C_6):S_4$, $C_2^2\times C_6\times S_4$, $C_2^2:D_6^2$, $S_4^2$, $C_6.C_2^3\times A_4$, $(C_2^2\times C_6).S_4$, $C_2^2\times A_4:C_{12}$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $A_4^2:C_4$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$

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

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

Order 288: $C_2\times C_6:S_4$ x 27, $C_3:\GL(2,\mathbb{Z}/4)$ x 21, $C_2^3:C_6^2$ x 11, $A_4\times C_3:D_4$ x 7, $C_2\times A_4\times D_6$ x 7, $C_2\times C_6.S_4$ x 6, $C_3\times \GL(2,\mathbb{Z}/4)$ x 6, $D_6:S_4$ x 6, $D_6\times S_4$ x 6, $C_2^3.C_6^2$ x 5, $C_2\times C_6\times S_4$ x 5, $C_{12}:S_4$ x 4, $C_{12}:S_4$ x 4, $C_2\times A_4^2$ x 4, $C_2^3.C_6^2$ x 3, $C_6:C_4\times A_4$ x 3, $D_6:S_4$ x 3, $D_6.S_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $A_4\times S_4$ x 3, $C_6^2:C_2^3$ x 3, $D_6^2:C_2$ x 2, $C_2\times A_4:C_{12}$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^5:C_3^2$ x 2, $\PSOPlus(4,3)$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $D_6^2:C_2$, $C_2^3.S_3^2$, $C_6^2:C_2^3$, $(C_2\times C_6):S_4$

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

Order 216: $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$, $C_3^2\times S_4$, $C_6^2:S_3$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_6.S_3^2$

Order 192: $C_2\times \GL(2,\mathbb{Z}/4)$ x 24, $D_6:C_2^4$ x 17, $C_2^4:D_6$ x 15, $C_2^4.D_6$ x 12, $C_2^4:D_6$ x 12, $C_2.\GL(2,\mathbb{Z}/4)$ x 11, $C_2^3\times S_4$ x 11, $C_2^4:D_6$ x 10, $C_2.\GL(2,\mathbb{Z}/4)$ x 9, $C_2^5:C_6$ x 9, $D_4\times S_4$ x 9, $C_2^4:D_6$ x 9, $C_2^5:C_6$ x 8, $C_2^4.D_6$ x 7, $C_2^4.D_6$ x 6, $C_2^3.D_{12}$ x 6, $C_2^3.D_{12}$ x 6, $C_2^3:D_{12}$ x 6, $C_2^4.D_6$ x 6, $C_2^4.D_6$ x 6, $C_2^3.D_{12}$ x 6, $C_2^4.D_6$ x 6, $A_4\times C_2^4$ x 4, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 4, $C_2^4:C_{12}$ x 3, $C_2^3.D_{12}$ x 3, $C_2^3.D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_{12}:C_2^4$ x 3, $C_{12}:C_2^4$ x 3, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3.S_4$ x 2, $C_2^3.D_{12}$, $C_2.\GL(2,\mathbb{Z}/4)$, $A_4:C_4^2$, $C_2^5\times C_6$, $D_6\times C_2^4$, $C_2^2\wr C_3$, $C_2^3:S_4$, $C_6.C_2^5$, $C_2^4:C_{12}$, $C_2^3.S_4$

Order 144: $C_6:S_4$ x 32, $C_2^2:C_6^2$ x 28, $C_6.S_4$ x 13, $A_4\times D_6$ x 12, $C_6\times S_4$ x 10, $C_6^2:C_2^2$ x 10, $C_6^2:C_2^2$ x 9, $C_6^2:C_2^2$ x 9, $C_{12}:D_6$ x 7, $D_6:D_6$ x 7, $C_6.D_{12}$ x 6, $C_6^2:C_4$ x 6, $C_6.D_{12}$ x 4, $S_3\times S_4$ x 4, $C_4:C_6^2$ x 3, $C_6:D_{12}$ x 3, $C_{12}:D_6$ x 3, $C_{12}\times A_4$ x 3, $D_6:D_6$ x 3, $A_4\times C_3:C_4$ x 3, $A_4:C_{12}$ x 3, $C_6^2:C_4$ x 3, $C_6.D_{12}$ x 2, $C_2^2\times C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $A_4^2$ x 2, $C_6^2:C_4$ x 2, $C_6:D_{12}$ x 2, $D_6.D_6$ x 2, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_2^4:C_3^2$, $D_6^2$

Order 128: $C_2^4:D_4$ x 6, $C_2^4:D_4$ x 6, $C_2\times D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2.D_4^2$ x 3, $C_2^4:D_4$ x 2, $C_2^3\wr C_2$ x 2, $D_4\times C_2^4$, $C_2^4:D_4$, $C_2^3.C_2^4$

Order 108: $C_3^2:D_6$ x 3, $C_3^2\times A_4$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 96: $\GL(2,\mathbb{Z}/4)$ x 18, $C_2^3\times A_4$ x 2, $C_2^4\times C_6$

Order 72: $C_3:S_4$ x 8, $C_6\times A_4$ x 2

Order 64: $C_2^6$

Order 48: $S_3\times D_4$ x 25, $C_2\times S_4$ x 10, $C_2^2\times A_4$ x 4, $C_6:D_4$ x 4, $C_2^3\times C_6$ x 3, $A_4:C_4$

Order 36: $C_6:S_3$ x 7, $S_3^2$ x 3, $C_3\times A_4$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$

Order 24: $C_3:D_4$ x 17, $S_4$ x 12, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2

Order 18: $C_3:S_3$ x 7

Order 16: $C_2\times D_4$ x 20, $C_2^4$ x 3

Order 12: $D_6$ x 23, $C_2\times C_6$ x 3, $A_4$ x 2

Order 8: $D_4$ x 39, $C_2^3$ x 2

Order 6: $S_3$ x 15, $C_6$

Order 4: $C_2^2$ x 17

Order 3: $C_3$

Order 2: $C_2$ x 12

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 6912: $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ ($C_1$)

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

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

Order 1152: $A_4\times \GL(2,\mathbb{Z}/4)$ ($S_3$) x 3, $C_2^3\times C_6:S_4$ ($S_3$), $C_2\times C_6\times \GL(2,\mathbb{Z}/4)$ ($S_3$)

Order 864: $A_4^2:S_3$ ($D_4$) x 2, $C_6\times A_4^2$ ($C_2^3$)

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

Order 432: $C_3\times A_4^2$ ($C_2\times D_4$)

Order 384: $C_2^2\times \GL(2,\mathbb{Z}/4)$ ($C_3:S_3$)

Order 288: $C_2\times A_4^2$ ($C_2\times D_6$) x 3, $C_2^3:C_6^2$ ($C_2\times D_6$) x 2, $C_2\times C_6:S_4$ ($C_3:D_4$) x 2, $C_3\times \GL(2,\mathbb{Z}/4)$ ($S_4$), $C_2\times C_6:S_4$ ($S_4$)

Order 192: $A_4\times C_2^4$ ($S_3^2$) x 3, $C_2^5\times C_6$ ($S_3^2$), $A_4\times C_2^4$ ($C_6:S_3$), $C_2^3\times S_4$ ($C_6:S_3$), $C_2^3.S_4$ ($C_6:S_3$)

Order 144: $A_4^2$ ($S_3\times D_4$) x 3, $C_2^2:C_6^2$ ($C_2\times S_4$) x 2, $C_6:S_4$ ($C_2\times S_4$) x 2, $C_2^2:C_6^2$ ($C_6:D_4$), $C_2^2:C_6^2$ ($S_3\times D_4$), $C_6\times S_4$ ($C_2\times S_4$), $A_4:C_{12}$ ($C_2\times S_4$)

Order 96: $C_2^3\times A_4$ ($S_3\times D_6$) x 3, $C_2^4\times C_6$ ($S_3\times D_6$), $C_2^3\times A_4$ ($C_6:D_6$), $\GL(2,\mathbb{Z}/4)$ ($C_3:S_4$)

Order 72: $C_6\times A_4$ ($C_2^2\times S_4$) x 2, $C_3:S_4$ ($\GL(2,\mathbb{Z}/4)$) x 2

Order 64: $C_2^6$ ($C_3:S_3^2$)

Order 48: $C_2^2\times A_4$ ($S_3\times S_4$) x 3, $C_2^2\times A_4$ ($D_6:D_6$) x 3, $C_2^3\times C_6$ ($S_3\times S_4$) x 2, $C_2^3\times C_6$ ($D_6:D_6$), $C_2^2\times A_4$ ($C_6:S_4$), $C_2^2\times A_4$ ($C_{12}:D_6$), $C_2\times S_4$ ($C_6:S_4$), $A_4:C_4$ ($C_6:S_4$)

Order 36: $C_3\times A_4$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_3\times A_4$ ($D_4\times S_4$)

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

Order 24: $C_2\times A_4$ ($D_6\times S_4$) x 3, $C_2^2\times C_6$ ($D_6\times S_4$) x 2, $C_2\times A_4$ ($C_2\times C_6:S_4$)

Order 16: $C_2^4$ ($A_4:S_3^2$), $C_2^4$ ($C_6^2:D_6$), $C_2^4$ ($C_6^2:D_6$)

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

Order 8: $C_2^3$ ($S_3\times C_6:S_4$), $C_2^3$ ($C_6:S_3\times S_4$)

Order 6: $C_6$ ($C_2\times S_4^2$)

Order 4: $C_2^2$ ($C_3:S_4^2$), $C_2^2$ ($C_6^2:(C_2\times S_4)$), $C_2^2$ ($C_3:S_3\times \GL(2,\mathbb{Z}/4)$)

Order 3: $C_3$ ($S_4\times \GL(2,\mathbb{Z}/4)$)

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

Order 1: $C_1$ ($C_3:S_4\times \GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 6912: $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ ($C_1$)

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

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

Order 1152: $C_2^3\times C_6:S_4$ ($S_3$), $C_2\times C_6\times \GL(2,\mathbb{Z}/4)$ ($S_3$), $A_4\times \GL(2,\mathbb{Z}/4)$ ($S_3$)

Order 864: $C_6\times A_4^2$ ($C_2^3$), $A_4^2:S_3$ ($D_4$)

Order 576: $C_2^4:C_6^2$ ($D_6$) x 2, $C_2^2\times C_6:S_4$ ($D_6$) x 2, $C_2^2\times C_6\times S_4$ ($D_6$), $C_2^2\times A_4^2$ ($D_6$), $A_4^2:C_2^2$ ($D_6$), $C_2^2\times A_4:C_{12}$ ($D_6$), $A_4^2:C_4$ ($D_6$)

Order 432: $C_3\times A_4^2$ ($C_2\times D_4$)

Order 384: $C_2^2\times \GL(2,\mathbb{Z}/4)$ ($C_3:S_3$)

Order 288: $C_2^3:C_6^2$ ($C_2\times D_6$) x 2, $C_3\times \GL(2,\mathbb{Z}/4)$ ($S_4$), $C_2\times C_6:S_4$ ($C_3:D_4$), $C_2\times C_6:S_4$ ($S_4$), $C_2\times A_4^2$ ($C_2\times D_6$)

Order 192: $C_2^5\times C_6$ ($S_3^2$), $A_4\times C_2^4$ ($C_6:S_3$), $A_4\times C_2^4$ ($S_3^2$), $C_2^3\times S_4$ ($C_6:S_3$), $C_2^3.S_4$ ($C_6:S_3$)

Order 144: $C_2^2:C_6^2$ ($C_2\times S_4$) x 2, $C_6:S_4$ ($C_2\times S_4$) x 2, $C_2^2:C_6^2$ ($C_6:D_4$), $C_2^2:C_6^2$ ($S_3\times D_4$), $C_6\times S_4$ ($C_2\times S_4$), $A_4^2$ ($S_3\times D_4$), $A_4:C_{12}$ ($C_2\times S_4$)

Order 96: $C_2^4\times C_6$ ($S_3\times D_6$), $C_2^3\times A_4$ ($C_6:D_6$), $C_2^3\times A_4$ ($S_3\times D_6$), $\GL(2,\mathbb{Z}/4)$ ($C_3:S_4$)

Order 72: $C_6\times A_4$ ($C_2^2\times S_4$) x 2, $C_3:S_4$ ($\GL(2,\mathbb{Z}/4)$)

Order 64: $C_2^6$ ($C_3:S_3^2$)

Order 48: $C_2^3\times C_6$ ($S_3\times S_4$) x 2, $C_2^3\times C_6$ ($D_6:D_6$), $C_2^2\times A_4$ ($C_6:S_4$), $C_2^2\times A_4$ ($S_3\times S_4$), $C_2^2\times A_4$ ($C_{12}:D_6$), $C_2^2\times A_4$ ($D_6:D_6$), $C_2\times S_4$ ($C_6:S_4$), $A_4:C_4$ ($C_6:S_4$)

Order 36: $C_3\times A_4$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_3\times A_4$ ($D_4\times S_4$)

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

Order 24: $C_2^2\times C_6$ ($D_6\times S_4$) x 2, $C_2\times A_4$ ($C_2\times C_6:S_4$), $C_2\times A_4$ ($D_6\times S_4$)

Order 16: $C_2^4$ ($A_4:S_3^2$), $C_2^4$ ($C_6^2:D_6$), $C_2^4$ ($C_6^2:D_6$)

Order 12: $C_2\times C_6$ ($S_4^2$), $C_2\times C_6$ ($C_2^4:S_3^2$), $C_2\times C_6$ ($S_3\times \GL(2,\mathbb{Z}/4)$), $A_4$ ($D_4\times C_3:S_4$), $A_4$ ($S_3\times \GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($S_3\times C_6:S_4$), $C_2^3$ ($C_6:S_3\times S_4$)

Order 6: $C_6$ ($C_2\times S_4^2$)

Order 4: $C_2^2$ ($C_3:S_4^2$), $C_2^2$ ($C_6^2:(C_2\times S_4)$), $C_2^2$ ($C_3:S_3\times \GL(2,\mathbb{Z}/4)$)

Order 3: $C_3$ ($S_4\times \GL(2,\mathbb{Z}/4)$)

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

Order 1: $C_1$ ($C_3:S_4\times \GL(2,\mathbb{Z}/4)$)

Series

Derived series

$C_3:S_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_3:S_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3:S_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_3:S_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2\times A_4^2:D_6$

$\rhd$

$C_2\times A_4^2:D_6$

$\rhd$

$C_2^6:C_3^3$

$\rhd$

$C_2^6:C_3^3$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$C_2^2:C_6^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3:S_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_3:S_4\times \GL(2,\mathbb{Z}/4)$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_6\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_3\times A_4^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 4 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 $126 \times 126$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $117 \times 117$ rational character table.