Abstract group 972.798: $C_3^3:S_3^2$

• Label: 972.798
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $36$
• Rank: $2$

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:	$C_3^4:(D_4\times D_6)$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	63	242	666	972
Conjugacy classes	1	3	31	16	51
Divisions	1	3	20	10	34
Autjugacy classes	1	2	14	7	24

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	12	18	15	2	0	4	0	51
Irr. rational chars.	4	10	9	2	6	2	1	34

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	10	10	10

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{3}=c^{6}=d^{3}=e^{3}=[a,e]=[b,c]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(12,13)(14,15), (2,4)(3,6)(5,7)(8,9)(10,11)(12,15)(13,14), (2,5,9)(4,7,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 61 \end{array}\right), \left(\begin{array}{rr} 31 & 10 \\ 60 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 9 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 71 & 30 \\ 45 & 1 \end{array}\right), \left(\begin{array}{rr} 73 & 54 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 46 & 45 \\ 45 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/90\Z)$
Transitive group:	36T1525	36T1545	36T1576		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $D_6$	$C_3^3$ $\,\rtimes\,$ $S_3^2$ (2)	$\He_3$ $\,\rtimes\,$ $S_3^2$	$(C_3^3:C_6)$ $\,\rtimes\,$ $S_3$ (2)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $(C_6\times S_3)$	$C_3^2$ . $(C_3\times S_3^2)$ (2)	$C_3$ . $(C_3\wr C_2^2)$	$(C_3^2:C_6)$ . $(C_3\times S_3)$	more information

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{90}\Z)$.

Homology

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

Subgroups

There are 3204 subgroups in 340 conjugacy classes, 35 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:S_3^2$
Commutator:	$G' \simeq$ $C_3^4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3\wr C_2^2$
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:	$\operatorname{soc} \simeq$ $C_3^3$	$G/\operatorname{soc} \simeq$ $C_6\times S_3$
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

Normal subgroups up to automorphism

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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

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

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

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

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

Order 162: $C_3^2\wr C_2$ x 5, $C_3^3:C_6$ x 4, $C_3^3:C_6$ x 4, $S_3\times \He_3$ x 3, $C_3^3:C_6$ x 2

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

Order 81: $C_3\times \He_3$ x 13, $C_3^4$ x 3

Order 54: $C_3^2:C_6$ x 18, $S_3\times C_3^2$ x 16, $C_3^2:C_6$ x 10, $C_3^2:S_3$ x 2, $C_2\times \He_3$

Order 36: $S_3^2$ x 5, $C_6\times S_3$ x 3, $C_6:S_3$

Order 27: $C_3^3$ x 34, $\He_3$ x 25

Order 18: $C_3\times S_3$ x 37, $C_3:S_3$ x 11, $C_3\times C_6$ x 5

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

Order 9: $C_3^2$ x 68

Order 6: $S_3$ x 13, $C_6$ x 10

Order 4: $C_2^2$

Order 3: $C_3$ x 20

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: $C_3^4:C_6$, $C_3^4:C_6$

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

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

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:C_6$ x 2, $S_3\times \He_3$ x 2, $C_3^3:C_6$ x 2, $C_3^3:C_6$

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

Order 81: $C_3\times \He_3$ x 8, $C_3^4$ x 3

Order 54: $C_3^2:C_6$ x 11, $S_3\times C_3^2$ x 9, $C_3^2:C_6$ x 4, $C_3^2:S_3$, $C_2\times \He_3$

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

Order 27: $C_3^3$ x 22, $\He_3$ x 11

Order 18: $C_3\times S_3$ x 19, $C_3:S_3$ x 6, $C_3\times C_6$ x 4

Order 12: $D_6$ x 3, $C_2\times C_6$

Order 9: $C_3^2$ x 38

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

Order 4: $C_2^2$

Order 3: $C_3$ x 14

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

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

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

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

Order 81: $C_3\times \He_3$ ($D_6$) x 2, $C_3^4$ ($C_2\times C_6$), $C_3^4$ ($D_6$)

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

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

Order 18: $C_3:S_3$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($C_3\times S_3^2$) x 3, $C_3^2$ ($C_3:S_3^2$), $C_3^2$ ($C_3^2:D_6$)

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

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: $C_3^4:C_6$ ($C_2$), $C_3^4:C_6$ ($C_2$)

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

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

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

Order 81: $C_3^4$ ($C_2\times C_6$), $C_3^4$ ($D_6$), $C_3\times \He_3$ ($D_6$)

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

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

Order 18: $C_3:S_3$ ($C_3^2:C_6$)

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

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

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

Series

Derived series

$C_3^3:S_3^2$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^3:S_3^2$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^2:C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3:S_3^2$

$\rhd$

$C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 12 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 $34 \times 34$ rational character table.