Abstract group 5184.od: $C_2^2\times S_3\wr S_3$

• Label: 5184.od
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_2^2\times S_3\wr S_3$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^3:C_2^2.D_6\times C_2^2:S_4$, of order \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	543	98	864	2670	144	432	432	5184
Conjugacy classes	1	23	4	8	44	1	4	3	88
Divisions	1	23	4	8	44	1	4	3	88
Autjugacy classes	1	7	4	2	13	1	1	1	30

Dimension	1	2	3	6	8	12	16
Irr. complex chars.	16	8	16	16	8	20	4	88
Irr. rational chars.	16	8	16	16	8	20	4	88

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	$1692553590$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=d^{6}=e^{6}=f^{6}=[c,e]=[e,f]=1, a^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(2,6,3,7,8,5)(4,9)(10,12)(11,13), (1,2,4,3)(5,7)(8,9)(10,11)(12,13), (1,3,7)(2,5,9)(4,8,6)(10,12)(11,13), (2,6,8,7,3,5)(4,9)(10,13)(11,12)\rangle$
Transitive group:	36T6138	36T6160			more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(S_3\wr S_3)$
Semidirect product:	$(S_3\times D_6^2)$ $\,\rtimes\,$ $S_3$	$(C_6^2:S_3)$ $\,\rtimes\,$ $S_4$	$(C_2\times S_3^3)$ $\,\rtimes\,$ $D_6$	$S_3^3$ $\,\rtimes\,$ $(C_2\times D_6)$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_3^3:(C_2\times \GL(2,3)))$	$\Aut(S_3^3:D_6)$	$\Aut((C_2\times S_3^3):C_6)$	$\Aut(C_3^3:\GL(2,\mathbb{Z}/4))$	all 19

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 113148 subgroups in 3812 conjugacy classes, 104 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3\wr S_3$
Commutator:	$G' \simeq$ $C_3^3:A_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times S_3\wr S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_2^2\times S_3\wr S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

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: $C_2^2\times S_3\wr S_3$

Order 2592: $S_3^3:D_6$ x 12, $C_2^2\times C_3^3:S_4$, $C_2^2\times C_3^3:S_4$, $C_2^2\times S_3\wr C_3$

Order 1728: $D_6^2:D_6$

Order 1296: $S_3\wr S_3$ x 16, $S_3^3:C_6$ x 6, $C_2\times C_3^3:S_4$ x 6, $C_2\times C_3^3:S_4$ x 6, $C_2^2\times C_3^3:A_4$, $C_6^2.S_3^2$

Order 864: $S_3^3:C_2^2$ x 24, $S_3\times D_6^2$ x 2, $D_6^2:S_3$ x 2, $C_6^2:D_{12}$, $D_6^2:C_6$, $C_6^2:C_4\times S_3$

Order 648: $C_2\times C_3^3:D_6$ x 12, $S_3\wr C_3$ x 4, $C_3^3:S_4$ x 4, $C_3^3:S_4$ x 4, $C_2\times C_3^3:A_4$ x 3, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $C_2^2\times C_3\wr S_3$

Order 576: $D_6^2:C_2^2$

Order 432: $S_3^3:C_2$ x 64, $C_2\times S_3^3$ x 30, $S_3^2:D_6$ x 24, $C_2\times C_3^2:D_{12}$ x 12, $C_6\times \SOPlus(4,2)$ x 12, $C_3^2:C_4\times D_6$ x 12, $C_3\times D_6^2$ x 3, $C_3:D_6^2$ x 2, $C_3:D_6^2$ x 2, $C_2^2\times C_3^3:C_4$, $C_6^2:C_{12}$, $C_6^2:D_6$

Order 324: $C_3^3:D_6$ x 16, $C_3^3:D_6$ x 6, $\He_3:D_6$ x 6, $C_3^3:D_6$ x 6, $C_3^2.C_6^2$, $C_3^3:A_4$

Order 288: $C_6^2:D_4$ x 28, $C_2\times D_6^2$ x 3, $C_2\times C_6^2:C_4$

Order 216: $S_3^3$ x 60, $S_3^2:S_3$ x 32, $C_6\times S_3^2$ x 30, $C_6:S_3^2$ x 24, $C_3^2:D_{12}$ x 16, $S_3^2:C_6$ x 16, $C_3^2:C_4\times S_3$ x 16, $C_6:S_3^2$ x 15, $C_6.S_3^2$ x 12, $C_2\times C_3^3:C_4$ x 6, $C_2\times C_3^2:C_{12}$ x 6, $C_6^2:C_6$ x 2, $S_3\times C_6^2$ x 2, $C_6^2:C_6$ x 2, $C_6^2:S_3$, $C_6^2:S_3$, $D_{18}:C_6$

Order 192: $C_2^3\times S_4$, $C_{12}:C_2^4$

Order 162: $C_3^3:S_3$ x 4, $C_3^3:C_6$ x 4, $C_3\wr S_3$ x 4, $C_3^3:C_6$ x 3

Order 144: $S_3^2:C_2^2$ x 112, $D_6^2$ x 72, $C_6^2:C_4$ x 14, $C_6^2:C_2^2$ x 4, $C_6^2:C_2^2$ x 2

Order 108: $C_3\times S_3^2$ x 36, $C_3:S_3^2$ x 32, $C_3^2:D_6$ x 16, $C_3:S_3^2$ x 15, $C_3^2:D_6$ x 12, $C_3^2\times D_6$ x 12, $C_3^2:D_6$ x 12, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 6, $C_{18}:C_6$ x 6, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 4, $C_3\times C_6^2$, $C_{18}:C_6$, $C_2^2\times \He_3$

Order 96: $D_4\times D_6$ x 24, $C_2^2\times S_4$ x 14, $C_2^3\times D_6$ x 3, $C_2^3:D_6$ x 2, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 81: $C_3\wr C_3$

Order 72: $S_3\times D_6$ x 276, $\SOPlus(4,2)$ x 64, $C_6\times D_6$ x 60, $C_6:D_6$ x 30, $C_2\times C_3^2:C_4$ x 28, $C_2\times C_6^2$ x 2, $C_2\times D_{18}$

Order 64: $D_4\times C_2^3$

Order 54: $C_3^2:C_6$ x 8, $C_3^2:C_6$ x 8, $S_3\times C_3^2$ x 8, $C_3^2:S_3$ x 4, $C_9:C_6$ x 4, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3, $C_9:C_6$ x 3, $C_2\times \He_3$ x 3

Order 48: $C_2^2\times D_6$ x 75, $S_3\times D_4$ x 64, $C_2\times S_4$ x 28, $C_6:D_4$ x 24, $C_6\times D_4$ x 12, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $C_2^2\times A_4$ x 7, $C_2^3\times C_6$ x 3, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 36: $S_3^2$ x 168, $C_6\times S_3$ x 136, $C_6:S_3$ x 68, $C_6^2$ x 16, $C_3^2:C_4$ x 8, $D_{18}$ x 6, $C_2\times C_{18}$

Order 32: $C_2^2\times D_4$ x 28, $C_2^5$ x 2, $C_2^3\times C_4$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2\times D_6$ x 330, $C_2^2\times C_6$ x 38, $C_3:D_4$ x 32, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3\times D_4$ x 16, $S_4$ x 8, $C_2\times A_4$ x 7, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 18: $C_3\times S_3$ x 48, $C_3:S_3$ x 24, $C_3\times C_6$ x 20, $D_9$ x 4, $C_{18}$ x 3

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 37, $C_2^2\times C_4$ x 14

Order 12: $D_6$ x 300, $C_2\times C_6$ x 88, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $C_2^3$ x 134, $D_4$ x 64, $C_2\times C_4$ x 28

Order 6: $S_3$ x 48, $C_6$ x 44

Order 4: $C_2^2$ x 114, $C_4$ x 8

Order 3: $C_3$ x 4

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $C_2^2\times S_3\wr S_3$

Order 2592: $C_2^2\times C_3^3:S_4$, $C_2^2\times C_3^3:S_4$, $C_2^2\times S_3\wr C_3$, $S_3^3:D_6$

Order 1728: $D_6^2:D_6$

Order 1296: $C_2^2\times C_3^3:A_4$, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_2\times C_3^3:S_4$, $S_3\wr S_3$, $C_6^2.S_3^2$

Order 864: $S_3\times D_6^2$ x 2, $D_6^2:S_3$ x 2, $S_3^3:C_2^2$ x 2, $C_6^2:D_{12}$, $D_6^2:C_6$, $C_6^2:C_4\times S_3$

Order 648: $C_2\times C_3^3:A_4$, $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:S_4$, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $C_2^2\times C_3\wr S_3$, $C_2\times C_3^3:D_6$

Order 576: $D_6^2:C_2^2$

Order 432: $C_2\times S_3^3$ x 4, $C_3\times D_6^2$ x 3, $S_3^2:D_6$ x 3, $C_3:D_6^2$ x 2, $C_3:D_6^2$ x 2, $C_2\times C_3^2:D_{12}$ x 2, $S_3^3:C_2$ x 2, $C_2^2\times C_3^3:C_4$, $C_6^2:C_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_6^2:D_6$

Order 324: $C_3^2.C_6^2$, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^3:A_4$

Order 288: $C_6^2:D_4$ x 6, $C_2\times D_6^2$ x 3, $C_2\times C_6^2:C_4$

Order 216: $C_6\times S_3^2$ x 5, $S_3^3$ x 5, $C_6:S_3^2$ x 4, $C_6:S_3^2$ x 3, $S_3^2:S_3$ x 3, $C_6^2:C_6$ x 2, $S_3\times C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$ x 2, $C_6^2:S_3$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$, $C_6^2:S_3$, $D_{18}:C_6$, $C_6.S_3^2$

Order 192: $C_2^3\times S_4$, $C_{12}:C_2^4$

Order 162: $C_3^3:C_6$, $C_3^3:S_3$, $C_3^3:C_6$, $C_3\wr S_3$

Order 144: $D_6^2$ x 19, $S_3^2:C_2^2$ x 8, $C_6^2:C_2^2$ x 4, $C_6^2:C_4$ x 3, $C_6^2:C_2^2$ x 2

Order 108: $C_3:S_3^2$ x 5, $C_3\times S_3^2$ x 5, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 3, $C_3^2\times D_6$ x 2, $C_3^2:D_6$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $C_3^2:D_6$, $C_{18}:C_6$, $C_3^2:D_6$

Order 96: $C_2^3\times D_6$ x 3, $C_2^2\times S_4$ x 3, $C_2^3:D_6$ x 2, $D_4\times D_6$ x 2, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 81: $C_3\wr C_3$

Order 72: $S_3\times D_6$ x 28, $C_6\times D_6$ x 18, $C_6:D_6$ x 9, $\SOPlus(4,2)$ x 6, $C_2\times C_3^2:C_4$ x 3, $C_2\times C_6^2$ x 2, $C_2\times D_{18}$

Order 64: $D_4\times C_2^3$

Order 54: $C_3^2:C_6$ x 3, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2:S_3$, $C_9:C_6$, $C_3^2\times C_6$, $C_3^2:S_3$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2^2\times D_6$ x 22, $C_2^3\times C_6$ x 3, $C_2\times S_4$ x 3, $C_6:D_4$ x 3, $C_2^2\times A_4$ x 2, $S_3\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_6\times D_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $C_4\times D_6$

Order 36: $S_3^2$ x 21, $C_6\times S_3$ x 20, $C_6:S_3$ x 10, $C_6^2$ x 6, $C_3^2:C_4$ x 2, $C_2\times C_{18}$, $D_{18}$

Order 32: $C_2^2\times D_4$ x 6, $C_2^5$ x 2, $C_2^3\times C_4$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2\times D_6$ x 44, $C_2^2\times C_6$ x 13, $C_3:D_4$ x 3, $D_{12}$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 14, $C_3:S_3$ x 7, $C_3\times C_6$ x 6, $C_{18}$, $D_9$

Order 16: $C_2^4$ x 11, $C_2\times D_4$ x 8, $C_2^2\times C_4$ x 3

Order 12: $D_6$ x 39, $C_2\times C_6$ x 18, $A_4$, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 4, $C_9$

Order 8: $C_2^3$ x 22, $D_4$ x 6, $C_2\times C_4$ x 3

Order 6: $S_3$ x 14, $C_6$ x 13

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $C_2^2\times S_3\wr S_3$ ($C_1$)

Order 2592: $S_3^3:D_6$ ($C_2$) x 12, $C_2^2\times C_3^3:S_4$ ($C_2$), $C_2^2\times C_3^3:S_4$ ($C_2$), $C_2^2\times S_3\wr C_3$ ($C_2$)

Order 1296: $S_3\wr S_3$ ($C_2^2$) x 16, $S_3^3:C_6$ ($C_2^2$) x 6, $C_2\times C_3^3:S_4$ ($C_2^2$) x 6, $C_2\times C_3^3:S_4$ ($C_2^2$) x 6, $C_2^2\times C_3^3:A_4$ ($C_2^2$)

Order 864: $S_3\times D_6^2$ ($S_3$)

Order 648: $S_3\wr C_3$ ($C_2^3$) x 4, $C_3^3:S_4$ ($C_2^3$) x 4, $C_3^3:S_4$ ($C_2^3$) x 4, $C_2\times C_3^3:A_4$ ($C_2^3$) x 3

Order 432: $C_2\times S_3^3$ ($D_6$) x 6, $C_3:D_6^2$ ($D_6$)

Order 324: $C_3^3:A_4$ ($C_2^4$)

Order 216: $S_3^3$ ($C_2\times D_6$) x 4, $C_6:S_3^2$ ($C_2\times D_6$) x 3, $C_6^2:S_3$ ($S_4$)

Order 108: $C_3^2:D_6$ ($C_2\times S_4$) x 6, $C_3\times C_6^2$ ($C_2\times S_4$), $C_3:S_3^2$ ($C_2^2\times D_6$)

Order 54: $C_3^2:S_3$ ($C_2^2\times S_4$) x 4, $C_3^2\times C_6$ ($C_2^2\times S_4$) x 3

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

Order 4: $C_2^2$ ($S_3\wr S_3$)

Order 2: $C_2$ ($S_3^3:D_6$) x 3

Order 1: $C_1$ ($C_2^2\times S_3\wr S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $C_2^2\times S_3\wr S_3$ ($C_1$)

Order 2592: $C_2^2\times C_3^3:S_4$ ($C_2$), $C_2^2\times C_3^3:S_4$ ($C_2$), $C_2^2\times S_3\wr C_3$ ($C_2$), $S_3^3:D_6$ ($C_2$)

Order 1296: $C_2^2\times C_3^3:A_4$ ($C_2^2$), $S_3^3:C_6$ ($C_2^2$), $C_2\times C_3^3:S_4$ ($C_2^2$), $C_2\times C_3^3:S_4$ ($C_2^2$), $S_3\wr S_3$ ($C_2^2$)

Order 864: $S_3\times D_6^2$ ($S_3$)

Order 648: $C_2\times C_3^3:A_4$ ($C_2^3$), $S_3\wr C_3$ ($C_2^3$), $C_3^3:S_4$ ($C_2^3$), $C_3^3:S_4$ ($C_2^3$)

Order 432: $C_3:D_6^2$ ($D_6$), $C_2\times S_3^3$ ($D_6$)

Order 324: $C_3^3:A_4$ ($C_2^4$)

Order 216: $C_6^2:S_3$ ($S_4$), $C_6:S_3^2$ ($C_2\times D_6$), $S_3^3$ ($C_2\times D_6$)

Order 108: $C_3\times C_6^2$ ($C_2\times S_4$), $C_3^2:D_6$ ($C_2\times S_4$), $C_3:S_3^2$ ($C_2^2\times D_6$)

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

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

Order 4: $C_2^2$ ($S_3\wr S_3$)

Order 2: $C_2$ ($S_3^3:D_6$)

Order 1: $C_1$ ($C_2^2\times S_3\wr S_3$)

Series

Derived series

$C_2^2\times S_3\wr S_3$

$\rhd$

$C_2^2\times S_3\wr S_3$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_2^2\times S_3\wr S_3$

$\rhd$

$C_2^2\times S_3\wr S_3$

$\rhd$

$C_2^2\times S_3\wr C_3$

$\rhd$

$C_2^2\times S_3\wr C_3$

$\rhd$

$S_3^3:C_6$

$\rhd$

$S_3^3:C_6$

$\rhd$

$S_3\wr C_3$

$\rhd$

$S_3\wr C_3$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^2\times S_3\wr S_3$

$\rhd$

$C_2^2\times S_3\wr S_3$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^3:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 12 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 $88 \times 88$ rational character table. Alternatively, you may search for characters of this group with desired properties.