Abstract group 864.4673: $S_4\times S_3^2$

• Label: 864.4673
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $10$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$S_4\times S_3^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	159	80	96	408	120	864
Conjugacy classes	1	11	7	4	17	5	45
Divisions	1	11	7	4	17	5	45
Autjugacy classes	1	8	5	3	10	3	30

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	8	12	8	6	8	1	2	45
Irr. rational chars.	8	12	8	6	8	1	2	45

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$80640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	7	7	7

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{6}=e^{6}=[a,b]=[a,c]=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $10$ $\langle(1,2)(3,5)(4,6), (1,2)(3,6)(4,5), (8,9), (8,9,10), (1,3,4)(2,5,6), (1,4,3)(2,5,6), (7,8)(9,10), (7,9)(8,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 23 & 12 \\ 36 & 47 \end{array}\right), \left(\begin{array}{rr} 31 & 45 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 36 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 30 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 0 \\ 30 & 31 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 59 & 50 \\ 45 & 19 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Transitive group:	24T2661	36T1363	36T1364	36T1365	all 6
Direct product:	$S_3$ ${}^2$ $\, \times\, $ $S_4$
Semidirect product:	$D_6^2$ $\,\rtimes\,$ $S_3$	$C_2^2$ $\,\rtimes\,$ $S_3^3$	$(S_3\times A_4)$ $\,\rtimes\,$ $D_6$ (2)	$A_4$ $\,\rtimes\,$ $(S_3\times D_6)$	all 27
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_2^2:C_6^2)$	$\Aut(C_6^2:C_6)$

Elements of the group are displayed as permutations of degree 10.

Homology

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

Subgroups

There are 6740 subgroups in 605 conjugacy classes, 48 normal (18 characteristic).

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

Special subgroups

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 864: $S_4\times S_3^2$

Order 432: $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $A_4\times S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$

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

Order 216: $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$, $S_3^3$

Order 144: $S_3\times S_4$ x 11, $D_6:D_6$ x 4, $D_6^2$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $C_{12}:D_6$ 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: $C_3:S_3^2$ x 3, $C_3\times S_3^2$ x 3, $C_3^2\times A_4$, $C_3:S_3^2$

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

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

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

Order 48: $S_3\times D_4$ x 17, $C_2\times S_4$ x 8, $C_2^2\times D_6$ 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 22, $C_6\times S_3$ x 14, $C_6:S_3$ x 7, $C_3\times A_4$ x 6, $C_3:C_{12}$ x 2, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 33, $C_3:D_4$ x 18, $D_{12}$ x 9, $C_4\times S_3$ x 9, $S_4$ x 9, $C_3\times D_4$ x 9, $C_2\times A_4$ x 5, $C_2^2\times C_6$ x 4, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 18: $C_3\times S_3$ x 20, $C_3:S_3$ x 11, $C_3\times C_6$ x 4

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

Order 9: $C_3^2$ x 7

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

Order 6: $S_3$ x 24, $C_6$ x 17

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $S_4\times S_3^2$

Order 432: $A_4\times S_3^2$, $A_4:S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$

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

Order 216: $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2\times S_4$, $S_3^3$

Order 144: $S_3\times S_4$ x 6, $D_6^2$ x 2, $D_6:D_6$ x 2, $D_6:D_6$ x 2, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $C_{12}:D_6$, $C_{12}:D_6$, $C_{12}:D_6$, $S_3\times D_{12}$, $C_4\times S_3^2$

Order 108: $C_3:S_3^2$ x 2, $C_3\times S_3^2$ x 2, $C_3^2\times A_4$, $C_3:S_3^2$

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

Order 72: $S_3\times D_6$ x 9, $C_3:S_4$ x 5, $S_3\times A_4$ x 4, $C_3\times S_4$ x 4, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6\times A_4$, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3\times D_{12}$, $S_3\times C_{12}$, $D_6:S_3$, $C_6.D_6$, $C_6.D_6$

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

Order 48: $S_3\times D_4$ x 9, $C_2\times S_4$ x 5, $C_2^2\times D_6$ x 2, $C_6:D_4$ x 2, $C_2^2\times A_4$, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $S_3^2$ x 14, $C_6\times S_3$ x 7, $C_6:S_3$ x 6, $C_3\times A_4$ x 4, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 18, $C_3:D_4$ x 10, $S_4$ x 6, $D_{12}$ x 5, $C_4\times S_3$ x 5, $C_3\times D_4$ x 5, $C_2\times A_4$ x 3, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 18: $C_3\times S_3$ x 11, $C_3:S_3$ x 8, $C_3\times C_6$ x 3

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 31, $C_2\times C_6$ x 10, $A_4$ x 3, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 5

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

Order 6: $S_3$ x 15, $C_6$ x 10

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $S_4\times S_3^2$ ($C_1$)

Order 432: $A_4:S_3^2$ ($C_2$) x 2, $C_6^2:D_6$ ($C_2$) x 2, $A_4\times S_3^2$ ($C_2$), $A_4:S_3^2$ ($C_2$), $C_6^2:D_6$ ($C_2$)

Order 216: $C_6^2:C_6$ ($C_2^2$) x 2, $C_3^2:S_4$ ($C_2^2$) x 2, $C_6^2:C_6$ ($C_2^2$), $C_3^2:S_4$ ($C_2^2$), $C_3^2\times S_4$ ($C_2^2$)

Order 144: $S_3\times S_4$ ($S_3$) x 2, $D_6^2$ ($S_3$)

Order 108: $C_3^2\times A_4$ ($C_2^3$)

Order 72: $C_6\times D_6$ ($D_6$) x 2, $S_3\times A_4$ ($D_6$) x 2, $C_3:S_4$ ($D_6$) x 2, $C_3\times S_4$ ($D_6$) x 2, $C_6:D_6$ ($D_6$)

Order 36: $C_3\times A_4$ ($C_2\times D_6$) x 2, $C_6^2$ ($C_2\times D_6$), $S_3^2$ ($S_4$)

Order 24: $C_2\times D_6$ ($S_3^2$) x 2, $S_4$ ($S_3^2$)

Order 18: $C_3\times S_3$ ($C_2\times S_4$) x 2, $C_3:S_3$ ($C_2\times S_4$)

Order 12: $C_2\times C_6$ ($S_3\times D_6$) x 2, $A_4$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($C_2^2\times S_4$)

Order 6: $S_3$ ($S_3\times S_4$) x 2

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

Order 3: $C_3$ ($D_6\times S_4$) x 2

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $S_4\times S_3^2$ ($C_1$)

Order 432: $A_4\times S_3^2$ ($C_2$), $A_4:S_3^2$ ($C_2$), $A_4:S_3^2$ ($C_2$), $C_6^2:D_6$ ($C_2$), $C_6^2:D_6$ ($C_2$)

Order 216: $C_6^2:C_6$ ($C_2^2$), $C_6^2:C_6$ ($C_2^2$), $C_3^2:S_4$ ($C_2^2$), $C_3^2:S_4$ ($C_2^2$), $C_3^2\times S_4$ ($C_2^2$)

Order 144: $D_6^2$ ($S_3$), $S_3\times S_4$ ($S_3$)

Order 108: $C_3^2\times A_4$ ($C_2^3$)

Order 72: $C_6:D_6$ ($D_6$), $C_6\times D_6$ ($D_6$), $S_3\times A_4$ ($D_6$), $C_3:S_4$ ($D_6$), $C_3\times S_4$ ($D_6$)

Order 36: $C_6^2$ ($C_2\times D_6$), $C_3\times A_4$ ($C_2\times D_6$), $S_3^2$ ($S_4$)

Order 24: $C_2\times D_6$ ($S_3^2$), $S_4$ ($S_3^2$)

Order 18: $C_3:S_3$ ($C_2\times S_4$), $C_3\times S_3$ ($C_2\times S_4$)

Order 12: $C_2\times C_6$ ($S_3\times D_6$), $A_4$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($C_2^2\times S_4$)

Order 6: $S_3$ ($S_3\times S_4$)

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

Order 3: $C_3$ ($D_6\times S_4$)

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

Series

Derived series

$S_4\times S_3^2$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$S_4\times S_3^2$

$\rhd$

$A_4\times S_3^2$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_4\times S_3^2$

$\rhd$

$C_3^2\times A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 43 larger groups in the database.

This group is a maximal quotient of 113 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 $45 \times 45$ rational character table. Alternatively, you may search for characters of this group with desired properties.