Abstract group 2592.lc: $C_3^2:D_6\times S_4$

• Label: 2592.lc
• Order: \( 2^{5} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^2:D_6\times S_4$
Order:	\(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\He_3.(D_4\times S_4)$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Composition factors:	$C_2$ x 5, $C_3$ x 4
Derived length:	$3$

This group is nonabelian, monomial (hence solvable), and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	279	242	168	1422	480	2592
Conjugacy classes	1	11	9	4	23	7	55
Divisions	1	11	9	4	23	7	55
Autjugacy classes	1	8	7	3	16	5	40

Dimension	1	2	3	4	6	8	12	18
Irr. complex chars.	8	12	8	6	12	1	4	4	55
Irr. rational chars.	8	12	8	6	12	1	4	4	55

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$725760$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	9	9	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{6}=c^{6}=d^{6}=e^{6}=[d,e]=1, a^{2}=c^{4}e^{4}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(2,5,8,3,7,6)(4,9)(10,12), (1,2,4,8,9,7)(3,6,5)(10,11,13), (1,3)(4,6)(5,9)(7,8)(11,13,12)\rangle$
Transitive group:	36T3967				more information
Direct product:	$S_4$ $\, \times\, $ $(C_3^2:D_6)$
Semidirect product:	$A_4$ $\,\rtimes\,$ $(C_6.S_3^2)$	$(C_6^2:D_6)$ $\,\rtimes\,$ $S_3$ (2)	$(C_6^2:D_6)$ $\,\rtimes\,$ $S_3$	$(C_3^2\times S_4)$ $\,\rtimes\,$ $D_6$ (2)	all 29
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times S_4)$ . $S_3^2$	$C_3$ . $(S_4\times S_3^2)$	$(C_2\times C_6)$ . $S_3^3$	$(C_3\times A_4)$ . $(S_3\times D_6)$	more information
Aut. group:	$\Aut(C_6^2:C_6)$	$\Aut(A_4\times C_3^2:C_6)$	$\Aut(C_6^2:S_3^2)$	$\Aut(C_6^2:S_3^2)$

Elements of the group are displayed as permutations of degree 13.

Homology

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

Subgroups

There are 23670 subgroups in 1089 conjugacy classes, 52 normal (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^2:D_6\times S_4$
Commutator:	$G' \simeq$ $A_4\times \He_3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_4\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2\times D_6$
Radical:	$R \simeq$ $C_3^2:D_6\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2592: $C_3^2:D_6\times S_4$

Order 1296: $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $A_4\times C_3^2:D_6$, $C_6^2:S_3^2$, $C_6^2:S_3^2$

Order 864: $S_4\times S_3^2$ x 2, $C_6.D_6^2$

Order 648: $A_4\times C_3^2:C_6$ x 2, $\He_3:S_4$ x 2, $A_4\times C_3^2:S_3$, $\He_3:S_4$, $S_4\times \He_3$, $C_3.S_3^3$

Order 432: $C_6^2:D_6$ x 5, $A_4:S_3^2$ x 4, $C_6^2:D_6$ x 4, $A_4\times S_3^2$ x 2, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_{12}.S_3^2$ x 2, $C_6^2:D_6$, $C_{12}.S_3^2$, $C_{12}.S_3^2$

Order 324: $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $A_4\times \He_3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^3:D_6$

Order 288: $D_6\times S_4$ x 3, $D_4\times S_3^2$ x 2

Order 216: $C_6.S_3^2$ x 11, $C_6^2:C_6$ x 5, $C_3^2:S_4$ x 5, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 3, $C_3^2\times S_4$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_4\times C_3^2:C_6$ x 2, $\He_3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $S_3^3$ x 2, $C_6^2:S_3$ x 2, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_6.S_3^2$

Order 162: $C_3^3:C_6$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^3:S_3$, $S_3\times \He_3$

Order 144: $S_3\times S_4$ x 17, $D_6:D_6$ x 8, $C_{12}:D_6$ x 5, $D_6^2$ x 4, $D_6:D_6$ x 4, $S_3\times D_{12}$ x 4, $A_4\times D_6$ x 3, $C_6:S_4$ x 3, $C_6\times S_4$ x 3, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $C_4\times S_3^2$ x 2

Order 108: $C_3^2:D_6$ x 13, $C_3^2:D_6$ x 10, $C_3\times S_3^2$ x 7, $C_3:S_3^2$ x 6, $C_3^2:D_6$ x 5, $C_3^2\times A_4$ x 3, $C_3^2:A_4$ x 3, $C_3^2:C_{12}$ x 2, $C_3:S_3^2$ x 2, $C_2^2\times \He_3$ x 2, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $D_4\times D_6$ x 3, $C_2^2\times S_4$

Order 81: $C_3\times \He_3$

Order 72: $S_3\times D_6$ x 25, $S_3\times A_4$ x 12, $C_3\times S_4$ x 12, $C_3:S_4$ x 11, $C_6\times D_6$ x 10, $C_6\wr C_2$ x 10, $C_3:D_{12}$ x 8, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $C_6:D_6$ x 4, $C_6^2:C_2$ x 4, $D_6:S_3$ x 4, $C_6.D_6$ x 4, $C_6\times A_4$ x 3, $D_4\times C_3^2$ x 3, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_6.D_6$ x 2

Order 54: $C_3^2:C_6$ x 10, $S_3\times C_3^2$ x 8, $C_3^2:C_6$ x 7, $C_3^2:S_3$ x 6, $C_3^2:S_3$ x 2, $C_2\times \He_3$ x 2

Order 48: $S_3\times D_4$ x 25, $C_2\times S_4$ x 9, $C_2^2\times D_6$ x 6, $C_6:D_4$ x 6, $C_6\times D_4$ x 3, $C_2\times D_{12}$ x 3, $C_4\times D_6$ x 3, $C_2^2\times A_4$

Order 36: $S_3^2$ x 39, $C_6\times S_3$ x 31, $C_6:S_3$ x 13, $C_3\times A_4$ x 10, $C_6^2$ x 6, $C_3:C_{12}$ x 5, $C_3\times C_{12}$ x 3, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$

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

Order 24: $C_2\times D_6$ x 48, $C_3:D_4$ x 26, $D_{12}$ x 13, $C_4\times S_3$ x 13, $C_3\times D_4$ x 13, $S_4$ x 11, $C_2^2\times C_6$ x 6, $C_2\times A_4$ x 6, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3

Order 18: $C_3\times S_3$ x 39, $C_3:S_3$ x 17, $C_3\times C_6$ x 9

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

Order 12: $D_6$ x 77, $C_2\times C_6$ x 24, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 5

Order 9: $C_3^2$ x 13

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $S_3$ x 32, $C_6$ x 23

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$ x 9

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2592: $C_3^2:D_6\times S_4$

Order 1296: $A_4\times C_3^2:D_6$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$

Order 864: $S_4\times S_3^2$, $C_6.D_6^2$

Order 648: $A_4\times C_3^2:S_3$, $A_4\times C_3^2:C_6$, $\He_3:S_4$, $\He_3:S_4$, $S_4\times \He_3$, $C_3.S_3^3$

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

Order 324: $A_4\times \He_3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^3:D_6$, $C_3^2:S_3^2$

Order 288: $D_6\times S_4$ x 2, $D_4\times S_3^2$

Order 216: $C_6.S_3^2$ x 8, $C_6^2:C_6$ x 3, $C_3^2:S_4$ x 3, $C_3^2:S_4$ x 2, $C_6^2:S_3$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2\times S_4$ x 2, $C_6^2:S_3$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:C_6$, $C_6.S_3^2$, $\He_3:D_4$, $C_6.S_3^2$, $C_6^2:C_6$, $C_3^2:S_4$, $S_3^3$

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

Order 144: $S_3\times S_4$ x 9, $D_6:D_6$ x 4, $C_{12}:D_6$ x 3, $D_6^2$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $D_6:D_6$ x 2, $S_3\times D_{12}$ x 2, $C_{12}:D_6$, $C_{12}:D_6$, $C_4\times S_3^2$

Order 108: $C_3^2:D_6$ x 9, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 5, $C_3\times S_3^2$ x 4, $C_3:S_3^2$ x 3, $C_3^2\times A_4$ x 2, $C_2^2\times \He_3$ x 2, $C_3^2:A_4$ x 2, $C_3^2:C_{12}$, $C_3:S_3^2$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $D_4\times D_6$ x 2, $C_2^2\times S_4$

Order 81: $C_3\times \He_3$

Order 72: $S_3\times D_6$ x 13, $C_3\times S_4$ x 8, $S_3\times A_4$ x 7, $C_6\times D_6$ x 6, $C_3:S_4$ x 6, $C_6\wr C_2$ x 6, $C_3:D_{12}$ x 4, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_6:D_6$ x 2, $C_6\times A_4$ x 2, $D_4\times C_3^2$ x 2, $C_6^2:C_2$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $C_3:D_{12}$, $C_{12}:S_3$, $C_6.D_6$

Order 54: $C_3^2:S_3$ x 5, $C_3^2:C_6$ x 5, $S_3\times C_3^2$ x 5, $C_3^2:C_6$ x 4, $C_2\times \He_3$ x 2, $C_3^2:S_3$

Order 48: $S_3\times D_4$ x 13, $C_2\times S_4$ x 6, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $C_6\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2, $C_2^2\times A_4$

Order 36: $S_3^2$ x 20, $C_6\times S_3$ x 19, $C_6:S_3$ x 7, $C_3\times A_4$ x 7, $C_6^2$ x 4, $C_3:C_{12}$ x 3, $C_3\times C_{12}$ x 2, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 27: $\He_3$ x 3, $C_3^3$ x 2

Order 24: $C_2\times D_6$ x 28, $C_3:D_4$ x 14, $C_3\times D_4$ x 9, $D_{12}$ x 7, $C_4\times S_3$ x 7, $S_4$ x 7, $C_2^2\times C_6$ x 4, $C_2\times A_4$ x 4, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 18: $C_3\times S_3$ x 24, $C_3:S_3$ x 9, $C_3\times C_6$ x 6

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

Order 12: $D_6$ x 45, $C_2\times C_6$ x 17, $C_{12}$ x 5, $A_4$ x 4, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 9

Order 8: $C_2^3$ x 12, $D_4$ x 10, $C_2\times C_4$ x 4

Order 6: $S_3$ x 19, $C_6$ x 16

Order 4: $C_2^2$ x 20, $C_4$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2592: $C_3^2:D_6\times S_4$ ($C_1$)

Order 1296: $C_6^2:S_3^2$ ($C_2$) x 2, $C_6^2:S_3^2$ ($C_2$) x 2, $A_4\times C_3^2:D_6$ ($C_2$), $C_6^2:S_3^2$ ($C_2$), $C_6^2:S_3^2$ ($C_2$)

Order 648: $A_4\times C_3^2:C_6$ ($C_2^2$) x 2, $\He_3:S_4$ ($C_2^2$) x 2, $A_4\times C_3^2:S_3$ ($C_2^2$), $\He_3:S_4$ ($C_2^2$), $S_4\times \He_3$ ($C_2^2$)

Order 432: $C_6^2:D_6$ ($S_3$) x 2, $C_6^2:D_6$ ($S_3$)

Order 324: $A_4\times \He_3$ ($C_2^3$)

Order 216: $C_6^2:C_6$ ($D_6$) x 2, $C_3^2:S_4$ ($D_6$) x 2, $C_3^2\times S_4$ ($D_6$) x 2, $C_6^2:C_6$ ($D_6$) x 2, $C_6^2:S_3$ ($D_6$)

Order 108: $C_3^2\times A_4$ ($C_2\times D_6$) x 2, $C_2^2\times \He_3$ ($C_2\times D_6$), $C_3^2:D_6$ ($S_4$)

Order 72: $C_6:D_6$ ($S_3^2$) x 2, $C_3\times S_4$ ($S_3^2$)

Order 54: $C_3^2:C_6$ ($C_2\times S_4$) x 2, $C_3^2:S_3$ ($C_2\times S_4$)

Order 36: $C_6^2$ ($S_3\times D_6$) x 2, $C_3\times A_4$ ($S_3\times D_6$)

Order 27: $\He_3$ ($C_2^2\times S_4$)

Order 24: $S_4$ ($C_3^2:D_6$)

Order 18: $C_3:S_3$ ($S_3\times S_4$) x 2

Order 12: $C_2\times C_6$ ($S_3^3$), $A_4$ ($C_6.S_3^2$)

Order 9: $C_3^2$ ($D_6\times S_4$) x 2

Order 4: $C_2^2$ ($C_3.S_3^3$)

Order 3: $C_3$ ($S_4\times S_3^2$)

Order 1: $C_1$ ($C_3^2:D_6\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2592: $C_3^2:D_6\times S_4$ ($C_1$)

Order 1296: $A_4\times C_3^2:D_6$ ($C_2$), $C_6^2:S_3^2$ ($C_2$), $C_6^2:S_3^2$ ($C_2$), $C_6^2:S_3^2$ ($C_2$), $C_6^2:S_3^2$ ($C_2$)

Order 648: $A_4\times C_3^2:S_3$ ($C_2^2$), $A_4\times C_3^2:C_6$ ($C_2^2$), $\He_3:S_4$ ($C_2^2$), $\He_3:S_4$ ($C_2^2$), $S_4\times \He_3$ ($C_2^2$)

Order 432: $C_6^2:D_6$ ($S_3$), $C_6^2:D_6$ ($S_3$)

Order 324: $A_4\times \He_3$ ($C_2^3$)

Order 216: $C_6^2:C_6$ ($D_6$), $C_3^2:S_4$ ($D_6$), $C_3^2\times S_4$ ($D_6$), $C_6^2:S_3$ ($D_6$), $C_6^2:C_6$ ($D_6$)

Order 108: $C_3^2\times A_4$ ($C_2\times D_6$), $C_2^2\times \He_3$ ($C_2\times D_6$), $C_3^2:D_6$ ($S_4$)

Order 72: $C_6:D_6$ ($S_3^2$), $C_3\times S_4$ ($S_3^2$)

Order 54: $C_3^2:S_3$ ($C_2\times S_4$), $C_3^2:C_6$ ($C_2\times S_4$)

Order 36: $C_6^2$ ($S_3\times D_6$), $C_3\times A_4$ ($S_3\times D_6$)

Order 27: $\He_3$ ($C_2^2\times S_4$)

Order 24: $S_4$ ($C_3^2:D_6$)

Order 18: $C_3:S_3$ ($S_3\times S_4$)

Order 12: $C_2\times C_6$ ($S_3^3$), $A_4$ ($C_6.S_3^2$)

Order 9: $C_3^2$ ($D_6\times S_4$)

Order 4: $C_2^2$ ($C_3.S_3^3$)

Order 3: $C_3$ ($S_4\times S_3^2$)

Order 1: $C_1$ ($C_3^2:D_6\times S_4$)

Series

Derived series

$C_3^2:D_6\times S_4$

$\rhd$

$A_4\times \He_3$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_3^2:D_6\times S_4$

$\rhd$

$C_6^2:S_3^2$

$\rhd$

$S_4\times \He_3$

$\rhd$

$A_4\times \He_3$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:D_6\times S_4$

$\rhd$

$A_4\times \He_3$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $55 \times 55$ rational character table. Alternatively, you may search for characters of this group with desired properties.