Abstract group 5184.qt: $C_3^2\times S_4^2$

• Label: 5184.qt
• Order: \( 2^{6} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $144$
• Rank: $2$

Group information

Description:	$C_3^2\times S_4^2$
Order:	\(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^5.C_6^2.D_6.C_2^2$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 4
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	99	728	156	2088	2112	5184
Conjugacy classes	1	8	35	7	100	74	225
Divisions	1	8	19	7	52	38	125
Autjugacy classes	1	5	5	4	9	6	30

Dimension	1	2	3	4	6	8	9	12	18
Irr. complex chars.	36	36	72	9	36	0	36	0	0	225
Irr. rational chars.	4	20	8	17	36	4	4	16	16	125

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$144$
Rank:	$2$
Inequivalent generating pairs:	$27$

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:	${\langle a, b, c, d, e, f \mid a^{6}=b^{6}=c^{2}=d^{6}=e^{6}=f^{2}=[b,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(5,7,6)(10,12), (4,5,7), (1,3,2), (4,5,6,7)(9,11,14), (1,2,3)(4,7,5,6)(8,13) \!\cdots\! \rangle$
Direct product:	$C_3$ ${}^2$ $\, \times\, $ $S_4$ ${}^2$
Semidirect product:	$A_4^2$ $\,\rtimes\,$ $C_6^2$	$(S_4\times C_6^2)$ $\,\rtimes\,$ $S_3$	$C_6^2$ $\,\rtimes\,$ $(S_3\times S_4)$	$A_4$ $\,\rtimes\,$ $(A_4:C_6^2)$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_2^5.C_2^5:D_{10})$

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

Homology

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

Subgroups

There are 25840 subgroups in 2116 conjugacy classes, 102 normal (10 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 5184: $C_3^2\times S_4^2$

Order 2592: $C_3^2\times A_4\times S_4$ x 2, $C_3^2\times \PSOPlus(4,3)$

Order 1728: $C_3\times S_4^2$ x 4, $D_4\times C_3^2\times S_4$ x 2

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

Order 864: $A_4^2:C_6$ x 16, $S_4\times C_6^2$ x 4, $A_4^2:C_6$ x 4, $C_3^2\times \GL(2,\mathbb{Z}/4)$ x 4, $D_4\times C_3^2\times A_4$ x 2, $C_6^2:D_{12}$ x 2, $C_3\times C_{12}\times S_4$ x 2, $C_6^2:S_4$

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

Order 576: $\GL(2,\mathbb{Z}/4):C_6$ x 8, $S_4^2$, $C_3^2\times D_4^2$

Order 432: $C_3\times A_4^2$ x 16, $A_4:C_6^2$ x 10, $C_6^2:D_6$ x 8, $A_4\times C_6^2$ x 4, $C_{12}:C_6^2$ x 2, $C_6^2:C_{12}$ x 2, $C_3\times A_4:C_{12}$ x 2, $C_6^2:A_4$

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

Order 288: $C_2^3.C_6^2$ x 16, $C_3\times \GL(2,\mathbb{Z}/4)$ x 16, $C_2\times C_6\times S_4$ x 16, $A_4\times S_4$ x 10, $C_{12}:S_4$ x 8, $C_{12}\times S_4$ x 8, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 4, $C_3\times C_2^2:S_4$ x 4, $D_4\times C_6^2$ x 4, $C_{12}^2:C_2$ x 2, $C_{12}^2:C_2$, $\PSOPlus(4,3)$

Order 216: $C_3^2\times S_4$ x 25, $C_6^2:C_6$ x 16, $C_3^2:S_4$ x 8, $S_3\times C_6^2$ x 4, $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 4, $D_4\times C_3^3$ x 2, $C_3^2\times D_{12}$ x 2, $C_6.C_6^2$ x 2

Order 192: $C_3\times D_4^2$ x 4, $D_4\times S_4$ x 2

Order 162: $S_3\times C_3^3$ x 2, $C_3^2\wr C_2$

Order 144: $C_6\times S_4$ x 40, $C_2^2:C_6^2$ x 32, $A_4^2$ x 25, $C_4:C_6^2$ x 24, $C_{12}\times A_4$ x 16, $C_2^4:C_3^2$ x 8, $C_{12}:D_6$ x 8, $A_4:C_{12}$ x 8, $C_6^2:C_4$ x 8, $C_2^2\times C_6^2$ x 4, $C_2\times C_6\times C_{12}$ x 4, $S_3\times S_4$ x 2, $C_{12}:C_{12}$ x 2, $C_{12}^2$

Order 108: $C_3^2\times A_4$ x 37, $C_3^2\times D_6$ x 10, $C_3\times C_6^2$ x 4, $C_3\times S_3^2$ x 4, $C_3^2\times C_{12}$ x 2, $C_3^2:C_{12}$ x 2

Order 96: $C_{12}:C_2^3$ x 16, $C_{12}:D_4$ x 16, $C_2^4:C_6$ x 16, $D_4\times A_4$ x 10, $D_4\times C_{12}$ x 8, $C_2^2\times S_4$ x 4, $\GL(2,\mathbb{Z}/4)$ x 4, $C_{12}:D_4$ x 4, $C_4:S_4$ x 2, $C_4\times S_4$ x 2, $C_2^2:S_4$

Order 81: $C_3^4$

Order 72: $C_3\times S_4$ x 46, $D_4\times C_3^2$ x 40, $C_6\times A_4$ x 32, $C_2\times C_6^2$ x 17, $C_6\times D_6$ x 16, $C_6\wr C_2$ x 16, $C_6\times C_{12}$ x 11, $S_3\times A_4$ x 10, $C_3\times D_{12}$ x 8, $S_3\times C_{12}$ x 8, $C_3:S_4$ x 2

Order 64: $D_4^2$

Order 54: $S_3\times C_3^2$ x 23, $C_3^2\times C_6$ x 4, $C_3^2:C_6$ x 4

Order 48: $C_6\times D_4$ x 96, $C_2^2:C_{12}$ x 32, $C_2^2\times A_4$ x 20, $C_2^3\times C_6$ x 16, $C_2^2\times C_{12}$ x 16, $C_2\times S_4$ x 10, $C_4\times A_4$ x 10, $C_4:C_{12}$ x 8, $C_2^2:A_4$ x 5, $C_4\times C_{12}$ x 4, $S_3\times D_4$ x 2, $A_4:C_4$ x 2

Order 36: $C_3\times A_4$ x 90, $C_6^2$ x 57, $C_6\times S_3$ x 40, $C_3\times C_{12}$ x 23, $C_3:C_{12}$ x 8, $S_3^2$

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 27: $C_3^3$ x 19

Order 24: $C_3\times D_4$ x 106, $C_2^2\times C_6$ x 68, $C_2\times C_{12}$ x 44, $C_2\times A_4$ x 20, $S_4$ x 9, $C_3:D_4$ x 4, $C_2\times D_6$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 18: $C_3\times C_6$ x 40, $C_3\times S_3$ x 38, $C_3:S_3$

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 120, $C_{12}$ x 38, $A_4$ x 25, $D_6$ x 10, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 50

Order 8: $D_4$ x 24, $C_2^3$ x 17, $C_2\times C_4$ x 11

Order 6: $C_6$ x 52, $S_3$ x 7

Order 4: $C_2^2$ x 25, $C_4$ x 7

Order 3: $C_3$ x 19

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 5184: $C_3^2\times S_4^2$

Order 2592: $C_3^2\times \PSOPlus(4,3)$, $C_3^2\times A_4\times S_4$

Order 1728: $C_3\times S_4^2$, $D_4\times C_3^2\times S_4$

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

Order 864: $S_4\times C_6^2$ x 2, $A_4^2:C_6$ x 2, $C_3^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_6^2:S_4$, $A_4^2:C_6$, $D_4\times C_3^2\times A_4$, $C_6^2:D_{12}$, $C_3\times C_{12}\times S_4$

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

Order 576: $S_4^2$, $\GL(2,\mathbb{Z}/4):C_6$, $C_3^2\times D_4^2$

Order 432: $A_4:C_6^2$ x 5, $C_3\times A_4^2$ x 3, $A_4\times C_6^2$ x 2, $C_6^2:A_4$, $C_6^2:D_6$, $C_{12}:C_6^2$, $C_6^2:C_{12}$, $C_3\times A_4:C_{12}$

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

Order 288: $C_6^2:D_4$ x 3, $C_2^3.C_6^2$ x 2, $C_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_6^2:D_4$ x 2, $C_2\times C_6\times S_4$ x 2, $A_4\times S_4$ x 2, $D_4\times C_6^2$ x 2, $C_{12}:S_4$, $C_{12}\times S_4$, $C_{12}^2:C_2$, $C_{12}^2:C_2$, $C_3\times C_2^2:S_4$, $\PSOPlus(4,3)$

Order 216: $C_3^2\times S_4$ x 7, $S_3\times C_6^2$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$, $D_4\times C_3^3$, $C_3^2\times D_{12}$, $C_6.C_6^2$

Order 192: $D_4\times S_4$, $C_3\times D_4^2$

Order 162: $C_3^2\wr C_2$, $S_3\times C_3^3$

Order 144: $C_4:C_6^2$ x 13, $C_6\times S_4$ x 5, $C_6^2:C_4$ x 5, $C_2^2:C_6^2$ x 4, $A_4^2$ x 4, $C_2^2\times C_6^2$ x 3, $C_2^4:C_3^2$ x 2, $C_2\times C_6\times C_{12}$ x 2, $C_{12}\times A_4$ x 2, $S_3\times S_4$, $C_{12}:D_6$, $A_4:C_{12}$, $C_{12}:C_{12}$, $C_{12}^2$

Order 108: $C_3^2\times A_4$ x 7, $C_3^2\times D_6$ x 5, $C_3\times C_6^2$ x 2, $C_3\times S_3^2$, $C_3^2\times C_{12}$, $C_3^2:C_{12}$

Order 96: $C_2^4:C_6$ x 3, $C_2^2\times S_4$ x 2, $C_{12}:C_2^3$ x 2, $D_4\times A_4$ x 2, $\GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:D_4$ x 2, $C_2^2:S_4$, $C_4:S_4$, $C_4\times S_4$, $C_{12}:D_4$, $D_4\times C_{12}$

Order 81: $C_3^4$

Order 72: $D_4\times C_3^2$ x 15, $C_2\times C_6^2$ x 10, $C_3\times S_4$ x 7, $C_6\times C_{12}$ x 6, $C_6\times A_4$ x 4, $C_6\times D_6$ x 2, $S_3\times A_4$ x 2, $C_6\wr C_2$ x 2, $C_3:S_4$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 64: $D_4^2$

Order 54: $S_3\times C_3^2$ x 6, $C_3^2\times C_6$ x 2, $C_3^2:C_6$

Order 48: $C_6\times D_4$ x 13, $C_2\times S_4$ x 5, $C_2^2:C_{12}$ x 5, $C_2^2\times A_4$ x 4, $C_2^3\times C_6$ x 3, $C_2^2:A_4$ x 2, $C_2^2\times C_{12}$ x 2, $C_4\times A_4$ x 2, $S_3\times D_4$, $A_4:C_4$, $C_4:C_{12}$, $C_4\times C_{12}$

Order 36: $C_6^2$ x 19, $C_3\times A_4$ x 11, $C_3\times C_{12}$ x 6, $C_6\times S_3$ x 5, $C_3:C_{12}$, $S_3^2$

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

Order 27: $C_3^3$ x 5

Order 24: $C_3\times D_4$ x 15, $C_2^2\times C_6$ x 10, $C_2\times C_{12}$ x 6, $S_4$ x 5, $C_2\times A_4$ x 4, $C_3:D_4$ x 2, $C_2\times D_6$ x 2, $D_{12}$, $C_4\times S_3$

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

Order 16: $C_2\times D_4$ x 13, $C_2^2:C_4$ x 5, $C_2^4$ x 3, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$

Order 12: $C_2\times C_6$ x 19, $A_4$ x 6, $C_{12}$ x 6, $D_6$ x 5, $C_3:C_4$

Order 9: $C_3^2$ x 9

Order 8: $D_4$ x 13, $C_2^3$ x 10, $C_2\times C_4$ x 6

Order 6: $C_6$ x 9, $S_3$ x 4

Order 4: $C_2^2$ x 15, $C_4$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 5184: $C_3^2\times S_4^2$ ($C_1$)

Order 2592: $C_3^2\times A_4\times S_4$ ($C_2$) x 2, $C_3^2\times \PSOPlus(4,3)$ ($C_2$)

Order 1728: $C_3\times S_4^2$ ($C_3$) x 4

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

Order 864: $A_4^2:C_6$ ($C_6$) x 8, $A_4^2:C_6$ ($C_6$) x 4, $S_4\times C_6^2$ ($S_3$) x 2

Order 576: $S_4^2$ ($C_3^2$)

Order 432: $C_3\times A_4^2$ ($C_2\times C_6$) x 4, $A_4\times C_6^2$ ($D_6$) x 2

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

Order 216: $C_3^2\times S_4$ ($S_4$) x 2

Order 144: $C_2^2:C_6^2$ ($C_6\times S_3$) x 8, $C_2^2\times C_6^2$ ($S_3^2$), $A_4^2$ ($C_6^2$)

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

Order 96: $C_2^2\times S_4$ ($S_3\times C_3^2$) x 2

Order 72: $C_3\times S_4$ ($C_3\times S_4$) x 8

Order 48: $C_2^3\times C_6$ ($C_3\times S_3^2$) x 4, $C_2^2\times A_4$ ($C_3^2\times D_6$) x 2

Order 36: $C_3\times A_4$ ($C_6\times S_4$) x 8, $C_6^2$ ($S_3\times S_4$) x 2

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

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

Order 12: $C_2\times C_6$ ($C_6^2:D_6$) x 8, $A_4$ ($A_4:C_6^2$) x 2

Order 9: $C_3^2$ ($S_4^2$)

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

Order 3: $C_3$ ($C_3\times S_4^2$) x 4

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 5184: $C_3^2\times S_4^2$ ($C_1$)

Order 2592: $C_3^2\times \PSOPlus(4,3)$ ($C_2$), $C_3^2\times A_4\times S_4$ ($C_2$)

Order 1728: $C_3\times S_4^2$ ($C_3$)

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

Order 864: $S_4\times C_6^2$ ($S_3$), $A_4^2:C_6$ ($C_6$), $A_4^2:C_6$ ($C_6$)

Order 576: $S_4^2$ ($C_3^2$)

Order 432: $A_4\times C_6^2$ ($D_6$), $C_3\times A_4^2$ ($C_2\times C_6$)

Order 288: $C_2\times C_6\times S_4$ ($C_3\times S_3$), $\PSOPlus(4,3)$ ($C_3\times C_6$), $A_4\times S_4$ ($C_3\times C_6$)

Order 216: $C_3^2\times S_4$ ($S_4$)

Order 144: $C_2^2\times C_6^2$ ($S_3^2$), $C_2^2:C_6^2$ ($C_6\times S_3$), $A_4^2$ ($C_6^2$)

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

Order 96: $C_2^2\times S_4$ ($S_3\times C_3^2$)

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

Order 48: $C_2^3\times C_6$ ($C_3\times S_3^2$), $C_2^2\times A_4$ ($C_3^2\times D_6$)

Order 36: $C_6^2$ ($S_3\times S_4$), $C_3\times A_4$ ($C_6\times S_4$)

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

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

Order 12: $C_2\times C_6$ ($C_6^2:D_6$), $A_4$ ($A_4:C_6^2$)

Order 9: $C_3^2$ ($S_4^2$)

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

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

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

Series

Derived series

$C_3^2\times S_4^2$

$\rhd$

$C_3^2\times S_4^2$

$\rhd$

$A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^2\times S_4^2$

$\rhd$

$C_3^2\times S_4^2$

$\rhd$

$C_3^2\times A_4\times S_4$

$\rhd$

$C_3^2\times A_4\times S_4$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$C_3\times A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^2\times S_4^2$

$\rhd$

$C_3^2\times S_4^2$

$\rhd$

$A_4^2$

$\rhd$

$A_4^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3^2$

$\lhd$

$C_3^2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $225 \times 225$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $125 \times 125$ rational character table.