Abstract group 972.795: $(C_3\times C_9):S_3^2$

• Label: 972.795
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{2} \)
• Perm deg.: $30$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$(C_3\times C_9):S_3^2$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$S_3\times C_3^3:\GL(2,3)$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
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	9	18
Elements	1	39	80	204	162	486	972
Conjugacy classes	1	3	13	13	28	32	90
Divisions	1	3	11	9	10	8	42
Autjugacy classes	1	3	5	6	4	5	24

Dimension	1	2	3	4	6	8	18	36
Irr. complex chars.	12	30	24	12	12	0	0	0	90
Irr. rational chars.	4	14	0	14	0	4	4	2	42

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$39312$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{3}=c^{6}=d^{3}=e^{9}=[a,d]=[a,e]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(29,30), (3,10)(4,12)(7,17)(8,19)(11,21)(14,22)(15,24)(18,26)(23,27), (1,2,6,5,9,16,13,20,25) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $(\He_3.C_6)$
Semidirect product:	$(C_3\times C_9)$ $\,\rtimes\,$ $S_3^2$	$(C_3^2\times C_9)$ $\,\rtimes\,$ $D_6$	$(C_9.C_3^2)$ $\,\rtimes\,$ $D_6$	$(C_3^2:C_{18})$ $\,\rtimes\,$ $S_3$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:S_3)$ . $C_6$	$(C_3^3:S_3)$ . $C_6$	$C_3^3$ . $(C_6\times S_3)$	$C_9$ . $(C_3:S_3^2)$	all 16
Aut. group:	$\Aut(C_{54}:C_6)$	$\Aut(C_3^3.C_{18})$

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 1452 subgroups in 292 conjugacy classes, 47 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_9$	$G/Z \simeq$ $C_3:S_3^2$
Commutator:	$G' \simeq$ $C_3\times \He_3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^2:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_9.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $(C_3\times C_9):S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9.C_3^3$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 972: $(C_3\times C_9):S_3^2$

Order 486: $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):S_3$, $S_3\times C_9.C_3^2$

Order 324: $C_9\times S_3^2$ x 4, $(C_3\times C_{18}):S_3$, $C_3^2:S_3^2$

Order 243: $C_9.C_3^3$

Order 162: $C_3^2:C_{18}$ x 8, $\He_3.C_6$ x 7, $C_3^2:C_{18}$ x 4, $C_3^3.C_6$ x 4, $C_6.C_3^3$, $C_3^3:S_3$, $C_3^3:S_3$, $S_3\times \He_3$

Order 108: $S_3\times C_{18}$ x 5, $C_3\times S_3^2$ x 4, $C_3^2:D_6$

Order 81: $C_9.C_3^2$ x 10, $C_9:C_3^2$ x 4, $C_3^2\times C_9$ x 4, $C_3\times \He_3$

Order 54: $S_3\times C_9$ x 22, $S_3\times C_3^2$ x 8, $C_3^2:S_3$ x 6, $C_3\times C_{18}$ x 5, $C_3^2:C_6$ x 4, $C_9:C_6$ x 4, $C_2\times \He_3$

Order 36: $C_6\times S_3$ x 5, $S_3^2$ x 4, $C_2\times C_{18}$

Order 27: $C_3\times C_9$ x 21, $C_9:C_3$ x 20, $\He_3$ x 5, $C_3^3$ x 4

Order 18: $C_3\times S_3$ x 26, $C_{18}$ x 8, $C_3\times C_6$ x 5, $C_3:S_3$ x 4

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

Order 9: $C_3^2$ x 17, $C_9$ x 10

Order 6: $S_3$ x 14, $C_6$ x 9

Order 4: $C_2^2$

Order 3: $C_3$ x 11

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $(C_3\times C_9):S_3^2$

Order 486: $(C_3^2\times C_9):S_3$, $(C_3^2\times C_9):S_3$, $S_3\times C_9.C_3^2$

Order 324: $(C_3\times C_{18}):S_3$, $C_3^2:S_3^2$, $C_9\times S_3^2$

Order 243: $C_9.C_3^3$

Order 162: $\He_3.C_6$ x 4, $C_3^2:C_{18}$ x 2, $C_6.C_3^3$, $C_3^3:S_3$, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$

Order 108: $S_3\times C_{18}$ x 2, $C_3\times S_3^2$, $C_3^2:D_6$

Order 81: $C_9.C_3^2$ x 4, $C_9:C_3^2$, $C_3\times \He_3$, $C_3^2\times C_9$

Order 54: $S_3\times C_9$ x 7, $C_3^2:S_3$ x 3, $C_3\times C_{18}$ x 2, $S_3\times C_3^2$ x 2, $C_3^2:C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 36: $C_6\times S_3$ x 2, $C_2\times C_{18}$, $S_3^2$

Order 27: $C_3\times C_9$ x 6, $C_9:C_3$ x 3, $\He_3$ x 2, $C_3^3$

Order 18: $C_3\times S_3$ x 8, $C_{18}$ x 5, $C_3\times C_6$ x 2, $C_3:S_3$

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

Order 9: $C_3^2$ x 5, $C_9$ x 4

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

Order 4: $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 972: $(C_3\times C_9):S_3^2$ ($C_1$)

Order 486: $(C_3^2\times C_9):S_3$ ($C_2$), $(C_3^2\times C_9):S_3$ ($C_2$), $S_3\times C_9.C_3^2$ ($C_2$)

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

Order 243: $C_9.C_3^3$ ($C_2^2$)

Order 162: $C_3^2:C_{18}$ ($S_3$) x 4, $C_3^3:S_3$ ($C_6$), $\He_3.C_6$ ($S_3$), $C_3^3:S_3$ ($C_6$), $S_3\times \He_3$ ($C_6$)

Order 81: $C_3^2\times C_9$ ($D_6$) x 4, $C_9.C_3^2$ ($D_6$), $C_3\times \He_3$ ($C_2\times C_6$)

Order 54: $S_3\times C_3^2$ ($C_3\times S_3$) x 4, $C_3^2:S_3$ ($C_3\times S_3$), $S_3\times C_9$ ($C_3:S_3$)

Order 27: $C_3^3$ ($C_6\times S_3$) x 4, $C_3\times C_9$ ($S_3^2$) x 4, $\He_3$ ($C_6\times S_3$), $C_3\times C_9$ ($C_6:S_3$)

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

Order 9: $C_3^2$ ($C_3\times S_3^2$) x 4, $C_3^2$ ($C_3^2:D_6$), $C_9$ ($C_3:S_3^2$)

Order 6: $S_3$ ($\He_3.C_6$)

Order 3: $C_3$ ($C_3^2:S_3^2$), $C_3$ ($(C_3\times C_{18}):S_3$)

Order 1: $C_1$ ($(C_3\times C_9):S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $(C_3\times C_9):S_3^2$ ($C_1$)

Order 486: $(C_3^2\times C_9):S_3$ ($C_2$), $(C_3^2\times C_9):S_3$ ($C_2$), $S_3\times C_9.C_3^2$ ($C_2$)

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

Order 243: $C_9.C_3^3$ ($C_2^2$)

Order 162: $C_3^3:S_3$ ($C_6$), $\He_3.C_6$ ($S_3$), $C_3^3:S_3$ ($C_6$), $S_3\times \He_3$ ($C_6$), $C_3^2:C_{18}$ ($S_3$)

Order 81: $C_9.C_3^2$ ($D_6$), $C_3\times \He_3$ ($C_2\times C_6$), $C_3^2\times C_9$ ($D_6$)

Order 54: $C_3^2:S_3$ ($C_3\times S_3$), $S_3\times C_9$ ($C_3:S_3$), $S_3\times C_3^2$ ($C_3\times S_3$)

Order 27: $C_3^3$ ($C_6\times S_3$), $\He_3$ ($C_6\times S_3$), $C_3\times C_9$ ($C_6:S_3$), $C_3\times C_9$ ($S_3^2$)

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

Order 9: $C_3^2$ ($C_3^2:D_6$), $C_3^2$ ($C_3\times S_3^2$), $C_9$ ($C_3:S_3^2$)

Order 6: $S_3$ ($\He_3.C_6$)

Order 3: $C_3$ ($C_3^2:S_3^2$), $C_3$ ($(C_3\times C_{18}):S_3$)

Order 1: $C_1$ ($(C_3\times C_9):S_3^2$)

Series

Derived series

$(C_3\times C_9):S_3^2$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$(C_3\times C_9):S_3^2$

$\rhd$

$S_3\times C_9.C_3^2$

$\rhd$

$C_3^2:C_{18}$

$\rhd$

$C_3^2\times C_9$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_9):S_3^2$

$\rhd$

$C_3\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_9$

Supergroups

This group is a maximal subgroup of 4 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 $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $42 \times 42$ rational character table.