Abstract group 1296.3492: $C_2\times C_3^3:S_4$

• Label: 1296.3492
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $11$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_2\times C_3^3:S_4$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$S_3^3:D_6$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	163	98	108	422	144	216	144	1296
Conjugacy classes	1	5	4	2	8	2	4	2	28
Divisions	1	5	4	2	8	2	2	2	26
Autjugacy classes	1	4	4	1	7	1	1	1	20

Dimension	1	2	3	6	8	12
Irr. complex chars.	4	2	4	8	6	4	28
Irr. rational chars.	4	2	4	4	6	6	26

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$117$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	6
Arbitrary	6	6	6

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{3}=c^{2}=d^{6}=e^{3}=f^{6}=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $11$ $\langle(2,4)(3,6)(5,8)(7,9), (10,11), (1,2,4)(3,5,7)(6,9,8), (3,6)(7,8), (3,6)(5,9), (2,5,9), (4,7,8), (1,3,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \\ 2 & 2 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 0 & 1 \\ 2 & 2 & 1 & 1 \\ 1 & 2 & 0 & 2 \\ 2 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 2 & 0 \\ 1 & 2 & 2 & 0 \\ 2 & 2 & 2 & 0 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 0 & 2 & 1 & 2 \\ 2 & 1 & 1 & 2 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 2 & 0 \\ 1 & 2 & 2 & 0 \\ 2 & 0 & 0 & 0 \\ 2 & 2 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 0 & 2 \\ 2 & 0 & 0 & 1 \\ 2 & 2 & 2 & 0 \\ 0 & 0 & 2 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	18T301	18T313	24T2878	24T2880	all 12
Direct product:	$C_2$ $\, \times\, $ $(C_3^3:S_4)$
Semidirect product:	$(C_6:S_3^2)$ $\,\rtimes\,$ $S_3$	$(C_3:S_3^2)$ $\,\rtimes\,$ $D_6$	$(C_3^2\times C_6)$ $\,\rtimes\,$ $S_4$	$C_3^3$ $\,\rtimes\,$ $(C_2\times S_4)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 4172 subgroups in 192 conjugacy classes, 11 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^3:S_4$
Commutator:	$G' \simeq$ $C_3^3:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times C_3^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_6$	$G/\operatorname{Fit} \simeq$ $S_4$
Radical:	$R \simeq$ $C_2\times C_3^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

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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Classes of subgroups up to conjugation

Order 1296: $C_2\times C_3^3:S_4$

Order 648: $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$

Order 432: $C_2\times C_3^2:D_{12}$

Order 324: $C_3^3:D_6$, $C_3^3:A_4$

Order 216: $C_3^2:D_{12}$ x 4, $C_6:S_3^2$ x 2, $C_2\times C_3^2:C_{12}$

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

Order 144: $S_3^2:C_2^2$

Order 108: $C_3:S_3^2$ x 5, $C_3^2:D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_{18}:C_6$ x 2, $C_3^2:D_6$

Order 81: $C_3\wr C_3$

Order 72: $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 3, $C_2\times C_3^2:C_4$

Order 54: $C_9:C_6$ x 4, $C_3^2:C_6$ x 4, $C_3^2:C_6$ x 2, $C_9:C_6$ x 2, $C_3^2\times C_6$, $C_2\times \He_3$

Order 48: $C_2\times S_4$, $C_2\times D_{12}$

Order 36: $S_3^2$ x 10, $C_6\times S_3$ x 5, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $D_{18}$ x 2

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

Order 24: $D_{12}$ x 4, $C_2\times D_6$ x 3, $S_4$ x 2, $C_2\times C_{12}$, $C_2\times A_4$

Order 18: $C_3\times S_3$ x 10, $C_3:S_3$ x 6, $C_3\times C_6$ x 4, $D_9$ x 4, $C_{18}$ x 2

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 16, $C_2\times C_6$ x 2, $C_{12}$ x 2, $A_4$

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

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

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

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_2\times C_3^3:S_4$

Order 648: $C_2\times C_3^3:A_4$, $C_3^3:S_4$

Order 432: $C_2\times C_3^2:D_{12}$

Order 324: $C_3^3:D_6$, $C_3^3:A_4$

Order 216: $C_6:S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $C_2\times C_3^2:C_{12}$

Order 162: $C_3^3:C_6$, $C_3^3:S_3$

Order 144: $S_3^2:C_2^2$

Order 108: $C_3:S_3^2$ x 4, $C_3^2:D_6$ x 2, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_3^2:D_6$

Order 81: $C_3\wr C_3$

Order 72: $S_3\times D_6$ x 3, $\SOPlus(4,2)$ x 2, $C_2\times C_3^2:C_4$

Order 54: $C_3^2:C_6$ x 3, $C_9:C_6$, $C_3^2:C_6$, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2\times S_4$, $C_2\times D_{12}$

Order 36: $S_3^2$ x 7, $C_6\times S_3$ x 5, $C_6:S_3$ x 3, $C_3^2:C_4$, $D_{18}$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2\times D_6$ x 3, $D_{12}$ x 2, $C_2\times C_{12}$, $C_2\times A_4$, $S_4$

Order 18: $C_3\times S_3$ x 7, $C_3\times C_6$ x 4, $C_3:S_3$ x 4, $C_{18}$, $D_9$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 13, $C_2\times C_6$ x 2, $A_4$, $C_{12}$

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

Order 8: $C_2^3$ x 2, $D_4$ x 2, $C_2\times C_4$

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 6, $C_4$

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_2\times C_3^3:S_4$ ($C_1$)

Order 648: $C_3^3:S_4$ ($C_2$) x 2, $C_2\times C_3^3:A_4$ ($C_2$)

Order 324: $C_3^3:A_4$ ($C_2^2$)

Order 216: $C_6:S_3^2$ ($S_3$)

Order 108: $C_3:S_3^2$ ($D_6$)

Order 54: $C_3^2\times C_6$ ($S_4$)

Order 27: $C_3^3$ ($C_2\times S_4$)

Order 2: $C_2$ ($C_3^3:S_4$)

Order 1: $C_1$ ($C_2\times C_3^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_2\times C_3^3:S_4$ ($C_1$)

Order 648: $C_2\times C_3^3:A_4$ ($C_2$), $C_3^3:S_4$ ($C_2$)

Order 324: $C_3^3:A_4$ ($C_2^2$)

Order 216: $C_6:S_3^2$ ($S_3$)

Order 108: $C_3:S_3^2$ ($D_6$)

Order 54: $C_3^2\times C_6$ ($S_4$)

Order 27: $C_3^3$ ($C_2\times S_4$)

Order 2: $C_2$ ($C_3^3:S_4$)

Order 1: $C_1$ ($C_2\times C_3^3:S_4$)

Series

Derived series

$C_2\times C_3^3:S_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^3:S_4$

$\rhd$

$C_3^3:S_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^3:S_4$

$\rhd$

$C_3^3:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

This group is a maximal quotient of 31 larger groups in the database.

Character theory

Complex character table

See the $28 \times 28$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $26 \times 26$ rational character table.