Abstract group 324.122: $C_3^2:S_3^2$

• Label: 324.122
• Order: \( 2^{2} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $18$
• Rank: $3$

Group information

Description:	$C_3^2:S_3^2$
Order:	\(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$S_3\times C_3^2:\GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Composition factors:	$C_2$ x 2, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	6
Elements	1	39	80	204	324
Conjugacy classes	1	3	13	13	30
Divisions	1	3	11	9	24
Autjugacy classes	1	3	5	6	15

Dimension	1	2	3	4	6	12
Irr. complex chars.	4	10	8	4	4	0	30
Irr. rational chars.	4	10	0	4	4	2	24

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$18$
Rank:	$3$
Inequivalent generating triples:	$4536$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{3}=c^{3}=d^{6}=e^{3}=[a,c]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(11,12), (2,4)(5,8)(7,9), (2,5,9)(4,7,8), (1,2,4)(3,5,8)(6,9,7), (10,11,12), (1,3,6)(2,5,9)(4,8,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 0 & 2 \\ 0 & 2 & 2 & 2 \\ 2 & 0 & 2 & 0 \\ 1 & 1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 2 & 2 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 1 & 0 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	18T135	27T118	36T507		more information
Direct product:	$S_3$ $\, \times\, $ $(C_3^2:S_3)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $D_6$	$\He_3$ $\,\rtimes\,$ $D_6$	$C_3^2$ $\,\rtimes\,$ $S_3^2$	$(S_3\times C_3^2)$ $\,\rtimes\,$ $S_3$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $(C_6:S_3)$	$C_3$ . $(C_3:S_3^2)$	$(C_3\times S_3)$ . $(C_3:S_3)$		more information
Aut. group:	$\Aut(C_{18}:C_6)$	$\Aut(C_3^3.C_6)$

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

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

Subgroups

There are 820 subgroups in 144 conjugacy classes, 25 normal (13 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

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

Classes of subgroups up to conjugation

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

Order 162: $C_3^3:S_3$, $C_3^3:S_3$, $S_3\times \He_3$

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

Order 81: $C_3\times \He_3$

Order 54: $S_3\times C_3^2$ x 8, $C_3^2:S_3$ x 6, $C_3^2:C_6$ x 4, $C_2\times \He_3$

Order 36: $C_6\times S_3$ x 5, $S_3^2$ x 4

Order 27: $\He_3$ x 5, $C_3^3$ x 4

Order 18: $C_3\times S_3$ x 26, $C_3\times C_6$ 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

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 324: $C_3^2:S_3^2$

Order 162: $C_3^3:S_3$, $C_3^3:S_3$, $S_3\times \He_3$

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

Order 81: $C_3\times \He_3$

Order 54: $C_3^2:S_3$ x 3, $S_3\times C_3^2$ x 2, $C_3^2:C_6$, $C_2\times \He_3$

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

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

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 2, $C_3:S_3$

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

Order 9: $C_3^2$ x 5

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 324: $C_3^2:S_3^2$ ($C_1$)

Order 162: $C_3^3:S_3$ ($C_2$), $C_3^3:S_3$ ($C_2$), $S_3\times \He_3$ ($C_2$)

Order 81: $C_3\times \He_3$ ($C_2^2$)

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

Order 27: $C_3^3$ ($D_6$) x 4, $\He_3$ ($D_6$)

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

Order 9: $C_3^2$ ($S_3^2$) x 4, $C_3^2$ ($C_6:S_3$)

Order 6: $S_3$ ($C_3^2:S_3$)

Order 3: $C_3$ ($C_3:S_3^2$), $C_3$ ($C_3^2:D_6$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 324: $C_3^2:S_3^2$ ($C_1$)

Order 162: $C_3^3:S_3$ ($C_2$), $C_3^3:S_3$ ($C_2$), $S_3\times \He_3$ ($C_2$)

Order 81: $C_3\times \He_3$ ($C_2^2$)

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

Order 27: $C_3^3$ ($D_6$), $\He_3$ ($D_6$)

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

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

Order 6: $S_3$ ($C_3^2:S_3$)

Order 3: $C_3$ ($C_3:S_3^2$), $C_3$ ($C_3^2:D_6$)

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

Series

Derived series

$C_3^2:S_3^2$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3^2:S_3^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:S_3^2$

$\rhd$

$C_3\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $30 \times 30$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $24 \times 24$ rational character table.