Abstract group 972.817: $C_3^3:S_3^2$

• Label: 972.817
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^3:S_3^2$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_3^4:(D_4\times \GL(2,3))$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6
Elements	1	63	242	666	972
Conjugacy classes	1	3	31	16	51
Divisions	1	3	23	11	38
Autjugacy classes	1	2	8	5	16

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	4	12	8	17	8	0	2	0	51
Irr. rational chars.	4	12	0	9	4	4	4	1	38

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$9072$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{3}=c^{3}=d^{6}=e^{3}=f^{3}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,4)(5,8)(7,9)(10,11)(12,14)(13,15), (2,4)(5,8)(7,9)(10,11)(12,15)(13,14) \!\cdots\! \rangle$
Transitive group:	36T1542				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $D_6$	$C_3^3$ $\,\rtimes\,$ $S_3^2$	$\He_3$ $\,\rtimes\,$ $S_3^2$	$(C_3^3:S_3)$ $\,\rtimes\,$ $S_3$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(C_6:S_3)$	$C_3^2$ . $(C_3:S_3^2)$	$C_3$ . $(C_3^2:S_3^2)$	$(C_3^2:C_6)$ . $(C_3:S_3)$	more information

Elements of the group are displayed as permutations of degree 15.

Homology

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

Subgroups

There are 3972 subgroups in 464 conjugacy classes, 40 normal (10 characteristic).

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

Special subgroups

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

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

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 972: $C_3^3:S_3^2$

Order 486: $C_3^4:S_3$ x 2, $C_3^4:C_6$

Order 324: $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 2

Order 243: $C_3^2\times \He_3$

Order 162: $C_3^2\wr C_2$ x 12, $C_3^3:S_3$ x 10, $S_3\times \He_3$ x 3, $C_3^3:S_3$ x 2

Order 108: $C_3\times S_3^2$ x 9, $C_3:S_3^2$ x 4, $C_3^2:D_6$

Order 81: $C_3\times \He_3$ x 15, $C_3^4$ x 4

Order 54: $S_3\times C_3^2$ x 30, $C_3^2:C_6$ x 29, $C_3^2:S_3$ x 10, $C_2\times \He_3$

Order 36: $S_3^2$ x 9, $C_6\times S_3$ x 6

Order 27: $C_3^3$ x 41, $\He_3$ x 25

Order 18: $C_3\times S_3$ x 77, $C_3:S_3$ x 9, $C_3\times C_6$ x 6

Order 12: $D_6$ x 6, $C_2\times C_6$

Order 9: $C_3^2$ x 83

Order 6: $S_3$ x 21, $C_6$ x 11

Order 4: $C_2^2$

Order 3: $C_3$ x 23

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $C_3^3:S_3^2$

Order 486: $C_3^4:S_3$, $C_3^4:C_6$

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

Order 243: $C_3^2\times \He_3$

Order 162: $C_3^2\wr C_2$ x 2, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2, $C_3^3:S_3$

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

Order 81: $C_3\times \He_3$ x 4, $C_3^4$

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

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

Order 27: $C_3^3$ x 8, $\He_3$ x 4

Order 18: $C_3\times S_3$ x 13, $C_3\times C_6$ x 2, $C_3:S_3$ x 2

Order 12: $D_6$ x 2, $C_2\times C_6$

Order 9: $C_3^2$ x 16

Order 6: $C_6$ x 5, $S_3$ x 5

Order 4: $C_2^2$

Order 3: $C_3$ x 8

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 972: $C_3^3:S_3^2$ ($C_1$)

Order 486: $C_3^4:S_3$ ($C_2$) x 2, $C_3^4:C_6$ ($C_2$)

Order 243: $C_3^2\times \He_3$ ($C_2^2$)

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

Order 81: $C_3^4$ ($D_6$) x 4, $C_3\times \He_3$ ($D_6$) x 2

Order 54: $C_3^2:C_6$ ($C_3:S_3$)

Order 27: $C_3^3$ ($S_3^2$) x 8, $C_3^3$ ($C_6:S_3$), $\He_3$ ($S_3^2$)

Order 18: $C_3:S_3$ ($C_3^2:S_3$)

Order 9: $C_3^2$ ($C_3:S_3^2$) x 4, $C_3^2$ ($C_3:S_3^2$) x 2, $C_3^2$ ($C_3^2:D_6$)

Order 3: $C_3$ ($C_3^2:S_3^2$) x 2, $C_3$ ($C_3^2:S_3^2$)

Order 1: $C_1$ ($C_3^3:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $C_3^3:S_3^2$ ($C_1$)

Order 486: $C_3^4:S_3$ ($C_2$), $C_3^4:C_6$ ($C_2$)

Order 243: $C_3^2\times \He_3$ ($C_2^2$)

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

Order 81: $C_3^4$ ($D_6$), $C_3\times \He_3$ ($D_6$)

Order 54: $C_3^2:C_6$ ($C_3:S_3$)

Order 27: $C_3^3$ ($C_6:S_3$), $C_3^3$ ($S_3^2$), $\He_3$ ($S_3^2$)

Order 18: $C_3:S_3$ ($C_3^2:S_3$)

Order 9: $C_3^2$ ($C_3:S_3^2$), $C_3^2$ ($C_3:S_3^2$), $C_3^2$ ($C_3^2:D_6$)

Order 3: $C_3$ ($C_3^2:S_3^2$), $C_3$ ($C_3^2:S_3^2$)

Order 1: $C_1$ ($C_3^3:S_3^2$)

Series

Derived series

$C_3^3:S_3^2$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3^3:S_3^2$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3:S_3^2$

$\rhd$

$C_3^2\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

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

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

Character theory

Complex character table

See the $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.