Abstract group 324.150: $C_3^3:D_6$

• Label: 324.150
• Order: \( 2^{2} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{3} \)
• Perm deg.: $14$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_3^3:D_6$
Order:	\(324\)\(\medspace = 2^{2} \cdot 3^{4} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_2\times C_3^3:C_3^2.Q_8.D_6$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
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	55	80	188	324
Conjugacy classes	1	3	17	21	42
Divisions	1	3	15	17	36
Autjugacy classes	1	2	4	5	12

Dimension	1	2	3	6	12
Irr. complex chars.	4	26	8	4	0	42
Irr. rational chars.	4	26	0	4	2	36

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	$110565$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{3}=c^{3}=d^{3}=e^{6}=[a,d]=[b,d]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,5)(3,6)(7,9)(11,12)(13,14), (13,14), (10,11,12), (1,2,5)(3,6,4)(7,9,8)(10,12,11), (1,3,6)(2,5,8)(4,7,9)(10,12,11), (1,4,8)(2,3,7)(5,6,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 0 & 0 & 2 & 2 \\ 1 & 0 & 1 & 0 \\ 2 & 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 2 & 2 \\ 0 & 2 & 0 & 0 \\ 0 & 1 & 1 & 2 \\ 0 & 2 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 2 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 2 \\ 1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	36T492				more information
Direct product:	$C_2$ $\, \times\, $ $(C_3^3:S_3)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $D_6$	$\He_3$ $\,\rtimes\,$ $D_6$	$(C_3^2\times C_6)$ $\,\rtimes\,$ $S_3$	$C_3^2$ $\,\rtimes\,$ $(C_6:S_3)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $(C_6:S_3)$	$C_6$ . $(C_3^2:S_3)$	$C_3$ . $(C_3^2:D_6)$	$(C_3\times C_6)$ . $(C_3:S_3)$	more information

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

Homology

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

Subgroups

There are 1228 subgroups in 240 conjugacy classes, 63 normal (11 characteristic).

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

Special subgroups

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

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Classes of subgroups up to automorphism

Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 324: $C_3^3:D_6$

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

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

Order 81: $C_3\times \He_3$

Order 54: $C_3^2:S_3$ x 18, $C_2\times \He_3$ x 9, $C_3^2:C_6$ x 8, $C_3^2\times C_6$ x 4

Order 36: $C_6\times S_3$ x 13, $C_6:S_3$ x 4

Order 27: $\He_3$ x 9, $C_3^3$ x 4

Order 18: $C_3\times S_3$ x 26, $C_3\times C_6$ x 21, $C_3:S_3$ x 8

Order 12: $D_6$ x 13, $C_2\times C_6$

Order 9: $C_3^2$ x 21

Order 6: $S_3$ x 26, $C_6$ x 17

Order 4: $C_2^2$

Order 3: $C_3$ x 15

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 324: $C_3^3:D_6$

Order 162: $C_6\times \He_3$, $C_3^3:S_3$

Order 108: $C_3^2:D_6$, $C_3^2:D_6$

Order 81: $C_3\times \He_3$

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

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

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

Order 18: $C_3\times C_6$ x 4, $C_3\times S_3$ x 2, $C_3:S_3$

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

Order 9: $C_3^2$ x 4

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 324: $C_3^3:D_6$ ($C_1$)

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

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

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

Order 27: $\He_3$ ($D_6$) x 9, $C_3^3$ ($D_6$) x 4

Order 18: $C_3\times C_6$ ($C_3:S_3$) x 13

Order 9: $C_3^2$ ($C_6:S_3$) x 13

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

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

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

Order 1: $C_1$ ($C_3^3:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 324: $C_3^3:D_6$ ($C_1$)

Order 162: $C_6\times \He_3$ ($C_2$), $C_3^3:S_3$ ($C_2$)

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

Order 54: $C_3^2\times C_6$ ($S_3$), $C_2\times \He_3$ ($S_3$)

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

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

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

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

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

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

Order 1: $C_1$ ($C_3^3:D_6$)

Series

Derived series

$C_3^3:D_6$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3^3:D_6$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3\times \He_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3:D_6$

$\rhd$

$C_3\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 34 larger groups in the database.

This group is a maximal quotient of 23 larger groups in the database.

Character theory

Complex character table

See the $42 \times 42$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.