Abstract group 1944.3941: $C_3^2:S_3^3$

• Label: 1944.3941
• Order: \( 2^{3} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_3^2:S_3^3$
Order:	\(1944\)\(\medspace = 2^{3} \cdot 3^{5} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$S_3^5.D_4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
Composition factors:	$C_2$ x 3, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	207	242	1494	1944
Conjugacy classes	1	7	41	29	78
Divisions	1	7	33	27	68
Autjugacy classes	1	3	11	8	23

Dimension	1	2	4	8	16
Irr. complex chars.	8	20	28	22	0	78
Irr. rational chars.	8	20	20	14	6	68

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$21504$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{3}=d^{6}=e^{3}=f^{3}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(3,6)(4,5)(8,9)(10,11)(12,14)(13,15), (3,4)(5,6), (1,2)(3,6)(4,5)(8,9) \!\cdots\! \rangle$
Transitive group:	24T4966	36T2826	36T2905	36T2942	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $S_3^3$ (2)	$C_3^5$ $\,\rtimes\,$ $C_2^3$	$(C_3^3:S_3)$ $\,\rtimes\,$ $D_6$	$C_3^3$ $\,\rtimes\,$ $(S_3\times D_6)$ (3)	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 21896 subgroups in 1524 conjugacy classes, 79 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^2:S_3^3$
Commutator:	$G' \simeq$ $C_3^5$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^2:S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $C_3^2:S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^5$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5$

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 1944: $C_3^2:S_3^3$

Order 972: $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$

Order 648: $C_3:S_3^3$ x 4, $C_3:S_3^3$

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

Order 324: $C_3^2:S_3^2$ x 16, $C_3^2:S_3^2$ x 14, $C_3\wr C_2^2$ x 12, $C_3^2:S_3^2$ x 3

Order 243: $C_3^5$

Order 216: $S_3^3$ x 8, $C_6:S_3^2$ x 2

Order 162: $C_3^2\wr C_2$ x 42, $C_3^3:C_6$ x 28, $S_3\times C_3^3$ x 6, $C_3^3:S_3$

Order 108: $C_3\times S_3^2$ x 38, $C_3:S_3^2$ x 32, $C_3:S_3^2$ x 30, $C_3^2:D_6$ x 6

Order 81: $C_3^4$ x 33

Order 72: $S_3\times D_6$ x 10

Order 54: $C_3^2:C_6$ x 152, $S_3\times C_3^2$ x 74, $C_3^2:S_3$ x 20, $C_3^2\times C_6$ x 2

Order 36: $S_3^2$ x 70, $C_6\times S_3$ x 26, $C_6:S_3$ x 10

Order 27: $C_3^3$ x 222

Order 24: $C_2\times D_6$ x 5

Order 18: $C_3\times S_3$ x 154, $C_3:S_3$ x 76, $C_3\times C_6$ x 18

Order 12: $D_6$ x 40, $C_2\times C_6$ x 5

Order 9: $C_3^2$ x 222

Order 8: $C_2^3$

Order 6: $S_3$ x 50, $C_6$ x 27

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 33

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1944: $C_3^2:S_3^3$

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

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

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

Order 324: $C_3^2:S_3^2$ x 3, $C_3\wr C_2^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 2

Order 243: $C_3^5$

Order 216: $S_3^3$ x 2, $C_6:S_3^2$

Order 162: $C_3^2\wr C_2$ x 10, $C_3^3:C_6$ x 7, $S_3\times C_3^3$ x 2, $C_3^3:S_3$

Order 108: $C_3:S_3^2$ x 7, $C_3\times S_3^2$ x 7, $C_3:S_3^2$ x 6, $C_3^2:D_6$ x 2

Order 81: $C_3^4$ x 11

Order 72: $S_3\times D_6$ x 3

Order 54: $C_3^2:C_6$ x 28, $S_3\times C_3^2$ x 15, $C_3^2:S_3$ x 6, $C_3^2\times C_6$

Order 36: $S_3^2$ x 13, $C_6\times S_3$ x 6, $C_6:S_3$ x 3

Order 27: $C_3^3$ x 43

Order 24: $C_2\times D_6$ x 2

Order 18: $C_3\times S_3$ x 27, $C_3:S_3$ x 17, $C_3\times C_6$ x 6

Order 12: $D_6$ x 9, $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 43

Order 8: $C_2^3$

Order 6: $S_3$ x 12, $C_6$ x 8

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 11

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1944: $C_3^2:S_3^3$ ($C_1$)

Order 972: $C_3^3:S_3^2$ ($C_2$) x 4, $C_3^3:S_3^2$ ($C_2$) x 2, $C_3^3:S_3^2$ ($C_2$)

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

Order 324: $C_3^2:S_3^2$ ($S_3$) x 4, $C_3^2:S_3^2$ ($S_3$)

Order 243: $C_3^5$ ($C_2^3$)

Order 162: $C_3^3:C_6$ ($D_6$) x 8, $C_3^2\wr C_2$ ($D_6$) x 6, $C_3^3:S_3$ ($D_6$)

Order 81: $C_3^4$ ($C_2\times D_6$) x 5

Order 54: $C_3^2:C_6$ ($S_3^2$) x 6, $C_3^2:S_3$ ($S_3^2$) x 4

Order 27: $C_3^3$ ($S_3\times D_6$) x 10

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

Order 9: $C_3^2$ ($S_3^3$) x 8, $C_3^2$ ($C_6:S_3^2$) x 2

Order 3: $C_3$ ($C_3:S_3^3$) x 4, $C_3$ ($C_3:S_3^3$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 1944: $C_3^2:S_3^3$ ($C_1$)

Order 972: $C_3^3:S_3^2$ ($C_2$), $C_3^3:S_3^2$ ($C_2$), $C_3^3:S_3^2$ ($C_2$)

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

Order 324: $C_3^2:S_3^2$ ($S_3$), $C_3^2:S_3^2$ ($S_3$)

Order 243: $C_3^5$ ($C_2^3$)

Order 162: $C_3^2\wr C_2$ ($D_6$) x 2, $C_3^3:S_3$ ($D_6$), $C_3^3:C_6$ ($D_6$)

Order 81: $C_3^4$ ($C_2\times D_6$) x 2

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

Order 27: $C_3^3$ ($S_3\times D_6$) x 3

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

Order 9: $C_3^2$ ($S_3^3$) x 2, $C_3^2$ ($C_6:S_3^2$)

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

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

Series

Derived series

$C_3^2:S_3^3$

$\rhd$

$C_3^5$

$\rhd$

$C_1$

Chief series

$C_3^2:S_3^3$

$\rhd$

$C_3^3:S_3^2$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:S_3^3$

$\rhd$

$C_3^5$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 18 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $78 \times 78$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.