Abstract group 5184.hx: $(S_3\times C_6^2):S_4$

• Label: 5184.hx
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$(S_3\times C_6^2):S_4$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_2^2\times C_6^2).D_6^2$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	171	890	468	1854	1800	5184
Conjugacy classes	1	6	13	5	55	22	102
Divisions	1	6	9	5	33	12	66
Autjugacy classes	1	6	9	5	29	11	61

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	12	12	12	3	38	0	25	0	102
Irr. rational chars.	4	8	4	5	14	1	21	9	66

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$144$

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:	${\langle a, b, c, d, e, f \mid a^{6}=b^{6}=c^{6}=d^{2}=e^{6}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,3)(2,5)(4,7)(8,9)(10,12,11)(14,15,17,19)(16,18), (1,2,4,6,8,5)(7,9)(10,11)(13,14)(15,16)(17,18)(19,20)\rangle$
Transitive group:	36T5603	36T5604			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^2:S_4)$ $\,\rtimes\,$ $S_3$	$(S_3\times C_6^2)$ $\,\rtimes\,$ $S_4$	$(C_6^2:S_4)$ $\,\rtimes\,$ $C_6$	$(C_6^2:S_4)$ $\,\rtimes\,$ $C_6$	all 23
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $(S_3\times S_4)$	$C_6^2$ . $(C_6\times S_4)$	$(C_6\times D_6)$ . $(C_3\times S_4)$	$C_2^2$ . $(C_6^2:S_3^2)$	all 12
Aut. group:	$\Aut(C_6^3.C_2^3)$	$\Aut(C_6^3.C_2^3)$

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 23940 subgroups in 1127 conjugacy classes, 38 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(S_3\times C_6^2):S_4$
Commutator:	$G' \simeq$ $C_6^2:A_4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_2\times C_6$	$G/\Phi \simeq$ $C_6^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $(S_3\times C_6^2):S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_6\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times \He_3$

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 5184: $(S_3\times C_6^2):S_4$

Order 2592: $C_2^4.C_3^3:S_3$, $C_3\times C_6^2:S_4$, $(C_2\times C_6^3):C_6$

Order 1728: $(C_6\times D_6):S_4$, $C_6^3.C_2^3$, $(C_2\times C_6^2):S_4$

Order 1296: $C_3\times C_6^2:A_4$, $C_6^2:S_3^2$

Order 864: $C_6^2:S_4$ x 4, $C_6^3.C_2^2$ x 2, $(C_6\times D_6):A_4$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:S_4$, $C_6^2:S_4$, $C_6^3:C_2^2$, $C_6^3.C_2^2$, $C_6^2.D_{12}$, $(C_2\times C_6^2):A_4$

Order 648: $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $S_3\times C_3^2:A_4$

Order 576: $C_3\times C_2^4:D_6$ x 2, $(C_2^2\times C_6):S_4$, $C_2^4:S_3^2$, $C_2^4:S_3^2$

Order 432: $C_6^2:A_4$ x 4, $D_6:C_6^2$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:A_4$ x 2, $C_6^3:C_2$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2:C_{12}$ x 2, $C_2\times C_6^3$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2:D_6$

Order 324: $C_3^3:A_4$ x 3, $C_3^2:S_3^2$

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

Order 216: $C_3^2:S_4$ x 10, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 4, $C_6^3$ x 3, $C_6.C_6^2$ x 3, $S_3\times C_6^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_6^2.C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$, $C_6\times S_3^2$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6^2:C_6$

Order 192: $C_2^4:D_6$ x 2, $C_2^3:S_4$, $C_3\times C_2\wr C_2^2$

Order 162: $C_3^3:C_6$, $S_3\times \He_3$, $C_3^3:C_6$

Order 144: $C_6^2:C_2^2$ x 19, $C_6^2:C_4$ x 17, $C_2^4:C_3^2$ x 8, $C_2^2\times C_6^2$ x 5, $C_{12}.D_6$ x 4, $C_{12}:D_6$ x 4, $C_4:C_6^2$ x 3, $C_6^2:C_4$ x 3, $C_6^2:C_2^2$ x 2, $D_{12}:C_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_6^2:C_4$ x 2, $C_6:S_4$, $S_3\times S_4$, $D_6:D_6$, $D_6.D_6$

Order 108: $C_3^2:A_4$ x 16, $C_3\times C_6^2$ x 8, $C_3^2\times A_4$ x 6, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^2:D_6$

Order 96: $C_2^4:C_6$ x 6, $C_2^4:S_3$ x 5, $C_2^2.D_{12}$ x 4, $C_2^3:C_{12}$ x 3, $C_2^2:S_4$ x 3, $C_2^3:A_4$ x 2, $C_2^3.D_6$ x 2, $D_4:D_6$ x 2, $C_6.C_2^4$

Order 81: $C_3\times \He_3$

Order 72: $C_6\wr C_2$ x 38, $C_2\times C_6^2$ x 27, $C_6:C_{12}$ x 19, $C_3:S_4$ x 12, $C_6\times D_6$ x 10, $D_4\times C_3^2$ x 6, $C_6^2:C_2$ x 4, $S_3\times C_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6.D_6$ x 2, $C_3\times D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3:D_{12}$ x 2, $C_6.D_6$ x 2, $C_6:D_6$, $S_3\times D_6$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$

Order 64: $C_2\wr C_2^2$

Order 54: $C_3^2:C_6$ x 4, $S_3\times C_3^2$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 2, $C_3^2:S_3$, $C_2\times \He_3$

Order 48: $C_6\times D_4$ x 15, $C_6:D_4$ x 12, $C_2^2:C_{12}$ x 12, $C_6.D_4$ x 10, $C_2^3\times C_6$ x 5, $C_2^2:A_4$ x 4, $D_4:S_3$ x 4, $S_3\times D_4$ x 4, $D_4:C_6$ x 3, $D_{12}:C_2$ x 2, $C_2\times S_4$

Order 36: $C_6^2$ x 65, $C_3\times A_4$ x 30, $C_6\times S_3$ x 19, $C_3:C_{12}$ x 17, $C_3\times C_{12}$ x 3, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $S_3^2$ x 2

Order 32: $C_2^3:C_4$ x 3, $C_2^2\wr C_2$ x 3, $D_4:C_2^2$

Order 27: $\He_3$ x 4, $C_3^3$ x 3

Order 24: $C_2^2\times C_6$ x 32, $C_3\times D_4$ x 27, $C_3:D_4$ x 24, $C_2\times C_{12}$ x 15, $C_6:C_4$ x 12, $C_2\times D_6$ x 7, $S_4$ x 7, $C_4\times S_3$ x 4, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_2\times A_4$ x 2, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 29, $C_3\times S_3$ x 11, $C_3:S_3$ x 6

Order 16: $C_2\times D_4$ x 8, $C_2^2:C_4$ x 5, $D_4:C_2$ x 3, $C_2^4$

Order 12: $C_2\times C_6$ x 75, $A_4$ x 15, $D_6$ x 13, $C_{12}$ x 12, $C_3:C_4$ x 10

Order 9: $C_3^2$ x 13

Order 8: $D_4$ x 13, $C_2\times C_4$ x 8, $C_2^3$ x 7, $Q_8$

Order 6: $C_6$ x 33, $S_3$ x 8

Order 4: $C_2^2$ x 14, $C_4$ x 5

Order 3: $C_3$ x 9

Order 2: $C_2$ x 6

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $(S_3\times C_6^2):S_4$

Order 2592: $C_2^4.C_3^3:S_3$, $C_3\times C_6^2:S_4$, $(C_2\times C_6^3):C_6$

Order 1728: $(C_6\times D_6):S_4$, $C_6^3.C_2^3$, $(C_2\times C_6^2):S_4$

Order 1296: $C_3\times C_6^2:A_4$, $C_6^2:S_3^2$

Order 864: $C_6^2:S_4$ x 4, $C_6^3.C_2^2$ x 2, $(C_6\times D_6):A_4$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:S_4$, $C_6^2:S_4$, $C_6^3:C_2^2$, $C_6^3.C_2^2$, $C_6^2.D_{12}$, $(C_2\times C_6^2):A_4$

Order 648: $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $S_3\times C_3^2:A_4$

Order 576: $C_3\times C_2^4:D_6$ x 2, $(C_2^2\times C_6):S_4$, $C_2^4:S_3^2$, $C_2^4:S_3^2$

Order 432: $C_6^2:A_4$ x 4, $D_6:C_6^2$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:A_4$ x 2, $C_6^3:C_2$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2:C_{12}$ x 2, $C_2\times C_6^3$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2:D_6$

Order 324: $C_3^3:A_4$ x 3, $C_3^2:S_3^2$

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

Order 216: $C_3^2:S_4$ x 8, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 4, $C_6^3$ x 3, $C_6.C_6^2$ x 3, $S_3\times C_6^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2:S_4$ x 2, $C_6^2.C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$, $C_6\times S_3^2$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6^2:C_6$

Order 192: $C_2^4:D_6$ x 2, $C_2^3:S_4$, $C_3\times C_2\wr C_2^2$

Order 162: $C_3^3:C_6$, $S_3\times \He_3$, $C_3^3:C_6$

Order 144: $C_6^2:C_2^2$ x 17, $C_6^2:C_4$ x 15, $C_2^4:C_3^2$ x 8, $C_2^2\times C_6^2$ x 5, $C_{12}.D_6$ x 4, $C_{12}:D_6$ x 4, $C_4:C_6^2$ x 3, $C_6^2:C_4$ x 3, $C_6^2:C_2^2$ x 2, $D_{12}:C_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_6^2:C_4$ x 2, $C_6:S_4$, $S_3\times S_4$, $D_6:D_6$, $D_6.D_6$

Order 108: $C_3^2:A_4$ x 12, $C_3\times C_6^2$ x 8, $C_3^2\times A_4$ x 6, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^2:D_6$

Order 96: $C_2^4:C_6$ x 6, $C_2^4:S_3$ x 5, $C_2^2.D_{12}$ x 4, $C_2^3:C_{12}$ x 3, $C_2^2:S_4$ x 3, $C_2^3:A_4$ x 2, $C_2^3.D_6$ x 2, $D_4:D_6$ x 2, $C_6.C_2^4$

Order 81: $C_3\times \He_3$

Order 72: $C_6\wr C_2$ x 34, $C_2\times C_6^2$ x 23, $C_6:C_{12}$ x 17, $C_6\times D_6$ x 10, $C_3:S_4$ x 10, $D_4\times C_3^2$ x 6, $C_6^2:C_2$ x 4, $S_3\times C_{12}$ x 4, $C_6\times C_{12}$ x 3, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6.D_6$ x 2, $C_3\times D_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_3:D_{12}$ x 2, $C_6.D_6$ x 2, $C_6:D_6$, $S_3\times D_6$, $C_3^2:Q_8$, $D_6:S_3$, $C_6.D_6$

Order 64: $C_2\wr C_2^2$

Order 54: $C_3^2:C_6$ x 4, $S_3\times C_3^2$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 2, $C_3^2:S_3$, $C_2\times \He_3$

Order 48: $C_6\times D_4$ x 14, $C_6:D_4$ x 11, $C_2^2:C_{12}$ x 11, $C_6.D_4$ x 9, $C_2^3\times C_6$ x 5, $C_2^2:A_4$ x 4, $D_4:S_3$ x 4, $S_3\times D_4$ x 4, $D_4:C_6$ x 3, $D_{12}:C_2$ x 2, $C_2\times S_4$

Order 36: $C_6^2$ x 55, $C_3\times A_4$ x 24, $C_6\times S_3$ x 17, $C_3:C_{12}$ x 15, $C_3\times C_{12}$ x 3, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $S_3^2$ x 2

Order 32: $C_2^3:C_4$ x 3, $C_2^2\wr C_2$ x 3, $D_4:C_2^2$

Order 27: $\He_3$ x 4, $C_3^3$ x 3

Order 24: $C_2^2\times C_6$ x 28, $C_3\times D_4$ x 25, $C_3:D_4$ x 22, $C_2\times C_{12}$ x 14, $C_6:C_4$ x 11, $C_2\times D_6$ x 7, $S_4$ x 6, $C_4\times S_3$ x 4, $D_{12}$ x 2, $C_3:Q_8$ x 2, $C_2\times A_4$ x 2, $C_3\times Q_8$

Order 18: $C_3\times C_6$ x 25, $C_3\times S_3$ x 11, $C_3:S_3$ x 6

Order 16: $C_2\times D_4$ x 8, $C_2^2:C_4$ x 5, $D_4:C_2$ x 3, $C_2^4$

Order 12: $C_2\times C_6$ x 64, $D_6$ x 12, $A_4$ x 12, $C_{12}$ x 11, $C_3:C_4$ x 9

Order 9: $C_3^2$ x 13

Order 8: $D_4$ x 13, $C_2\times C_4$ x 8, $C_2^3$ x 7, $Q_8$

Order 6: $C_6$ x 29, $S_3$ x 8

Order 4: $C_2^2$ x 14, $C_4$ x 5

Order 3: $C_3$ x 9

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $(S_3\times C_6^2):S_4$ ($C_1$)

Order 2592: $C_2^4.C_3^3:S_3$ ($C_2$), $C_3\times C_6^2:S_4$ ($C_2$), $(C_2\times C_6^3):C_6$ ($C_2$)

Order 1728: $(C_6\times D_6):S_4$ ($C_3$)

Order 1296: $C_3\times C_6^2:A_4$ ($C_2^2$)

Order 864: $C_6^2:S_4$ ($C_6$), $C_6^2:S_4$ ($C_6$), $C_6^2:S_4$ ($S_3$), $C_6^3.C_2^2$ ($S_3$), $(C_6\times D_6):A_4$ ($C_6$)

Order 432: $C_2\times C_6^3$ ($D_6$), $C_6^2:A_4$ ($C_2\times C_6$), $C_6^2:A_4$ ($D_6$)

Order 288: $C_6^2:D_4$ ($C_3\times S_3$), $(C_2\times C_6):S_4$ ($C_3\times S_3$)

Order 216: $S_3\times C_6^2$ ($S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_6\times S_3$), $C_2^2\times C_6^2$ ($S_3^2$), $C_2^4:C_3^2$ ($C_6\times S_3$)

Order 108: $C_3\times C_6^2$ ($C_2\times S_4$)

Order 96: $C_2^4:S_3$ ($C_3^2:C_6$)

Order 72: $C_6\times D_6$ ($C_3\times S_4$)

Order 48: $C_2^3\times C_6$ ($C_3\times S_3^2$), $C_2^3\times C_6$ ($C_3^2:D_6$)

Order 36: $C_6^2$ ($C_6\times S_4$), $C_6^2$ ($S_3\times S_4$)

Order 27: $C_3^3$ ($C_2^3:S_4$)

Order 24: $C_2\times D_6$ ($C_3^2:S_4$)

Order 16: $C_2^4$ ($C_3^2:S_3^2$)

Order 12: $C_2\times C_6$ ($C_6^2:D_6$), $C_2\times C_6$ ($C_6^2:D_6$)

Order 9: $C_3^2$ ($C_3\times C_2^3:S_4$), $C_3^2$ ($C_2^4:S_3^2$)

Order 4: $C_2^2$ ($C_6^2:S_3^2$)

Order 3: $C_3$ ($C_3\times C_2^4:S_3^2$), $C_3$ ($(C_2\times C_6^2):S_4$)

Order 1: $C_1$ ($(S_3\times C_6^2):S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $(S_3\times C_6^2):S_4$ ($C_1$)

Order 2592: $C_2^4.C_3^3:S_3$ ($C_2$), $C_3\times C_6^2:S_4$ ($C_2$), $(C_2\times C_6^3):C_6$ ($C_2$)

Order 1728: $(C_6\times D_6):S_4$ ($C_3$)

Order 1296: $C_3\times C_6^2:A_4$ ($C_2^2$)

Order 864: $C_6^2:S_4$ ($C_6$), $C_6^2:S_4$ ($C_6$), $C_6^2:S_4$ ($S_3$), $C_6^3.C_2^2$ ($S_3$), $(C_6\times D_6):A_4$ ($C_6$)

Order 432: $C_2\times C_6^3$ ($D_6$), $C_6^2:A_4$ ($C_2\times C_6$), $C_6^2:A_4$ ($D_6$)

Order 288: $C_6^2:D_4$ ($C_3\times S_3$), $(C_2\times C_6):S_4$ ($C_3\times S_3$)

Order 216: $S_3\times C_6^2$ ($S_4$)

Order 144: $C_2^2\times C_6^2$ ($C_6\times S_3$), $C_2^2\times C_6^2$ ($S_3^2$), $C_2^4:C_3^2$ ($C_6\times S_3$)

Order 108: $C_3\times C_6^2$ ($C_2\times S_4$)

Order 96: $C_2^4:S_3$ ($C_3^2:C_6$)

Order 72: $C_6\times D_6$ ($C_3\times S_4$)

Order 48: $C_2^3\times C_6$ ($C_3\times S_3^2$), $C_2^3\times C_6$ ($C_3^2:D_6$)

Order 36: $C_6^2$ ($C_6\times S_4$), $C_6^2$ ($S_3\times S_4$)

Order 27: $C_3^3$ ($C_2^3:S_4$)

Order 24: $C_2\times D_6$ ($C_3^2:S_4$)

Order 16: $C_2^4$ ($C_3^2:S_3^2$)

Order 12: $C_2\times C_6$ ($C_6^2:D_6$), $C_2\times C_6$ ($C_6^2:D_6$)

Order 9: $C_3^2$ ($C_3\times C_2^3:S_4$), $C_3^2$ ($C_2^4:S_3^2$)

Order 4: $C_2^2$ ($C_6^2:S_3^2$)

Order 3: $C_3$ ($C_3\times C_2^4:S_3^2$), $C_3$ ($(C_2\times C_6^2):S_4$)

Order 1: $C_1$ ($(S_3\times C_6^2):S_4$)

Series

Derived series

$(S_3\times C_6^2):S_4$

$\rhd$

$(S_3\times C_6^2):S_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(S_3\times C_6^2):S_4$

$\rhd$

$(S_3\times C_6^2):S_4$

$\rhd$

$C_2^4.C_3^3:S_3$

$\rhd$

$C_2^4.C_3^3:S_3$

$\rhd$

$C_3\times C_6^2:A_4$

$\rhd$

$C_3\times C_6^2:A_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(S_3\times C_6^2):S_4$

$\rhd$

$(S_3\times C_6^2):S_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_6^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $66 \times 66$ rational character table.