Abstract group 1296.3538: $S_3^4$

• Label: 1296.3538
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$S_3^4$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$S_3\wr S_4$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.

Group statistics

Order	1	2	3	6
Elements	1	255	80	960	1296
Conjugacy classes	1	15	15	50	81
Divisions	1	15	15	50	81
Autjugacy classes	1	4	4	6	15

Dimension	1	2	4	8	16
Irr. complex chars.	16	32	24	8	1	81
Irr. rational chars.	16	32	24	8	1	81

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	$23991240$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{6}=d^{6}=e^{3}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(7,8)(9,11)(10,12), (7,8)(9,12)(10,11), (1,2)(3,6)(4,5), (1,2)(3,5)(4,6), (1,3,4)(2,5,6), (7,9,10)(8,11,12), (1,4,3)(2,5,6), (7,10,9)(8,11,12)\rangle$
Transitive group:	24T2925	36T2230	36T2260		more information
Direct product:	$S_3$ ${}^4$
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $D_6^2$	$C_3^4$ $\,\rtimes\,$ $C_2^4$	$(C_3:S_3^2)$ $\,\rtimes\,$ $D_6$	$(C_3\times S_3^2)$ $\,\rtimes\,$ $D_6$	all 26
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 12.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 19244 subgroups in 1569 conjugacy classes, 170 normal (5 characteristic).

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

Special subgroups

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

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 1296: $S_3^4$

Order 648: $C_3:S_3^3$ x 6, $C_3:S_3^3$ x 4, $C_3\times S_3^3$ x 4, $C_3:S_3^3$

Order 432: $C_2\times S_3^3$ x 4

Order 324: $C_3^2:S_3^2$ x 12, $C_3^2:S_3^2$ x 6, $C_3^2\times S_3^2$ x 6, $C_3^2:S_3^2$ x 4, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 3

Order 216: $S_3^3$ x 38, $C_6:S_3^2$ x 12, $C_6\times S_3^2$ x 12, $C_6:S_3^2$ x 4

Order 162: $C_3^2\wr C_2$ x 6, $C_3^3:C_6$ x 4, $S_3\times C_3^3$ x 4, $C_3^3:S_3$

Order 144: $D_6^2$ x 6

Order 108: $C_3:S_3^2$ x 70, $C_3\times S_3^2$ x 66, $C_3:S_3^2$ x 22, $C_3^2:D_6$ x 12, $C_3^2\times D_6$ x 12, $C_3^2:D_6$ x 4

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 84, $C_6\times D_6$ x 12, $C_6:D_6$ x 6

Order 54: $C_3^2:C_6$ x 46, $S_3\times C_3^2$ x 46, $C_3^2:S_3$ x 19, $C_3^2\times C_6$ x 4

Order 48: $C_2^2\times D_6$ x 4

Order 36: $S_3^2$ x 151, $C_6\times S_3$ x 96, $C_6:S_3$ x 52, $C_6^2$ x 6

Order 27: $C_3^3$ x 15

Order 24: $C_2\times D_6$ x 62, $C_2^2\times C_6$ x 4

Order 18: $C_3\times S_3$ x 110, $C_3:S_3$ x 70, $C_3\times C_6$ x 34

Order 16: $C_2^4$

Order 12: $D_6$ x 152, $C_2\times C_6$ x 34

Order 9: $C_3^2$ x 36

Order 8: $C_2^3$ x 15

Order 6: $S_3$ x 65, $C_6$ x 50

Order 4: $C_2^2$ x 35

Order 3: $C_3$ x 15

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $S_3^4$

Order 648: $C_3:S_3^3$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3\times S_3^3$

Order 432: $C_2\times S_3^3$

Order 324: $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3\wr C_2^2$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$

Order 216: $S_3^3$ x 5, $C_6:S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $C_3^2\wr C_2$, $S_3\times C_3^3$

Order 144: $D_6^2$

Order 108: $C_3:S_3^2$ x 7, $C_3\times S_3^2$ x 5, $C_3:S_3^2$ x 3, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 7, $C_6:D_6$, $C_6\times D_6$

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

Order 48: $C_2^2\times D_6$

Order 36: $S_3^2$ x 13, $C_6:S_3$ x 6, $C_6\times S_3$ x 6, $C_6^2$

Order 27: $C_3^3$ x 4

Order 24: $C_2\times D_6$ x 7, $C_2^2\times C_6$

Order 18: $C_3:S_3$ x 11, $C_3\times S_3$ x 10, $C_3\times C_6$ x 4

Order 16: $C_2^4$

Order 12: $D_6$ x 14, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 4

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

Order 4: $C_2^2$ x 6

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $S_3^4$ ($C_1$)

Order 648: $C_3:S_3^3$ ($C_2$) x 6, $C_3:S_3^3$ ($C_2$) x 4, $C_3\times S_3^3$ ($C_2$) x 4, $C_3:S_3^3$ ($C_2$)

Order 324: $C_3^2:S_3^2$ ($C_2^2$) x 12, $C_3^2:S_3^2$ ($C_2^2$) x 6, $C_3^2\times S_3^2$ ($C_2^2$) x 6, $C_3^2:S_3^2$ ($C_2^2$) x 4, $C_3\wr C_2^2$ ($C_2^2$) x 4, $C_3^2:S_3^2$ ($C_2^2$) x 3

Order 216: $S_3^3$ ($S_3$) x 4

Order 162: $C_3^2\wr C_2$ ($C_2^3$) x 6, $C_3^3:C_6$ ($C_2^3$) x 4, $S_3\times C_3^3$ ($C_2^3$) x 4, $C_3^3:S_3$ ($C_2^3$)

Order 108: $C_3:S_3^2$ ($D_6$) x 12, $C_3\times S_3^2$ ($D_6$) x 12, $C_3:S_3^2$ ($D_6$) x 4

Order 81: $C_3^4$ ($C_2^4$)

Order 54: $C_3^2:C_6$ ($C_2\times D_6$) x 12, $S_3\times C_3^2$ ($C_2\times D_6$) x 12, $C_3^2:S_3$ ($C_2\times D_6$) x 4

Order 36: $S_3^2$ ($S_3^2$) x 6

Order 27: $C_3^3$ ($C_2^2\times D_6$) x 4

Order 18: $C_3\times S_3$ ($S_3\times D_6$) x 12, $C_3:S_3$ ($S_3\times D_6$) x 6

Order 9: $C_3^2$ ($D_6^2$) x 6

Order 6: $S_3$ ($S_3^3$) x 4

Order 3: $C_3$ ($C_2\times S_3^3$) x 4

Order 1: $C_1$ ($S_3^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $S_3^4$ ($C_1$)

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

Order 324: $C_3^2:S_3^2$ ($C_2^2$), $C_3^2:S_3^2$ ($C_2^2$), $C_3^2:S_3^2$ ($C_2^2$), $C_3\wr C_2^2$ ($C_2^2$), $C_3^2:S_3^2$ ($C_2^2$), $C_3^2\times S_3^2$ ($C_2^2$)

Order 216: $S_3^3$ ($S_3$)

Order 162: $C_3^3:S_3$ ($C_2^3$), $C_3^3:C_6$ ($C_2^3$), $C_3^2\wr C_2$ ($C_2^3$), $S_3\times C_3^3$ ($C_2^3$)

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

Order 81: $C_3^4$ ($C_2^4$)

Order 54: $C_3^2:S_3$ ($C_2\times D_6$), $C_3^2:C_6$ ($C_2\times D_6$), $S_3\times C_3^2$ ($C_2\times D_6$)

Order 36: $S_3^2$ ($S_3^2$)

Order 27: $C_3^3$ ($C_2^2\times D_6$)

Order 18: $C_3:S_3$ ($S_3\times D_6$), $C_3\times S_3$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($D_6^2$)

Order 6: $S_3$ ($S_3^3$)

Order 3: $C_3$ ($C_2\times S_3^3$)

Order 1: $C_1$ ($S_3^4$)

Series

Derived series

$S_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$S_3^4$

$\rhd$

$C_3:S_3^3$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$S_3^4$

$\rhd$

$C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 9 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 $81 \times 81$ rational character table. Alternatively, you may search for characters of this group with desired properties.