Abstract group 1296.3527: $(C_3\times S_3^2):D_6$

• Label: 1296.3527
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$(C_3\times S_3^2):D_6$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\PSU(3,2):C_2\times \AGL(2,3)$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	219	80	180	672	144	1296
Conjugacy classes	1	7	14	2	26	4	54
Divisions	1	7	14	2	26	4	54
Autjugacy classes	1	5	3	2	6	1	18

Dimension	1	2	4	8
Irr. complex chars.	8	18	12	16	54
Irr. rational chars.	8	18	12	16	54

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$6720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{12}=d^{3}=e^{3}=[a,d]=[a,e]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(8,9)(11,12), (4,5), (1,2)(3,5)(4,6), (3,6)(4,5), (1,3,6)(2,4,5), (1,3,6)(2,5,4), (7,8,9)(10,11,12), (7,9,8)(10,11,12)\rangle$
Transitive group:	36T2246				more information
Direct product:	$(C_3:S_3)$ $\, \times\, $ $\SOPlus(4,2)$
Semidirect product:	$(S_3^2:C_6)$ $\,\rtimes\,$ $S_3$	$S_3^2$ $\,\rtimes\,$ $(C_6:S_3)$	$(C_3^3:S_3)$ $\,\rtimes\,$ $D_4$	$(C_3\times S_3^2)$ $\,\rtimes\,$ $D_6$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3)$ . $(C_6:D_6)$	$(C_3^2:S_3^2)$ . $C_2^2$	$(C_3^2\wr C_2)$ . $C_2^3$	$(C_3^2:C_6)$ . $(C_2\times D_6)$	more information

Elements of the group are displayed as permutations of degree 12.

Homology

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

Subgroups

There are 12020 subgroups in 688 conjugacy classes, 56 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_3\times S_3^2):D_6$
Commutator:	$G' \simeq$ $C_3^2\wr C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $(C_3\times S_3^2):D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2\times D_4$
Radical:	$R \simeq$ $(C_3\times S_3^2):D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_2\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 1296: $(C_3\times S_3^2):D_6$

Order 648: $C_3:S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$

Order 432: $S_3^3:C_2$ x 4

Order 324: $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2\times S_3^2$ x 2, $C_3^2:S_3^2$, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 216: $S_3^3$ x 8, $S_3^2:S_3$ x 8, $C_3^2:D_{12}$ x 4, $S_3^2:C_6$ x 4, $C_3^2:C_4\times S_3$ x 4, $C_6:S_3^2$ x 2

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

Order 144: $S_3^2:C_2^2$, $C_{12}:D_6$

Order 108: $C_3:S_3^2$ x 28, $C_3\times S_3^2$ x 16, $C_3:S_3^2$ x 8, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 2, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 10, $\SOPlus(4,2)$ x 4, $C_6:D_6$ x 2, $C_6^2:C_2$ x 2, $C_2\times C_3^2:C_4$, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$

Order 54: $C_3^2:C_6$ x 24, $S_3\times C_3^2$ x 24, $C_3^2:S_3$ x 16, $C_3^2\times C_6$ x 2

Order 48: $S_3\times D_4$ x 4

Order 36: $S_3^2$ x 46, $C_6:S_3$ x 24, $C_6\times S_3$ x 18, $C_3^2:C_4$ x 2, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 27: $C_3^3$ x 14

Order 24: $C_2\times D_6$ x 10, $C_3:D_4$ x 8, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4

Order 18: $C_3:S_3$ x 52, $C_3\times S_3$ x 52, $C_3\times C_6$ x 20

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 48, $C_2\times C_6$ x 10, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 32

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

Order 6: $S_3$ x 40, $C_6$ x 26

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 14

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $(C_3\times S_3^2):D_6$

Order 648: $C_3:S_3^3$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$

Order 432: $S_3^3:C_2$

Order 324: $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 216: $C_6:S_3^2$, $S_3^3$, $C_3^2:D_{12}$, $S_3^2:S_3$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

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

Order 144: $S_3^2:C_2^2$, $C_{12}:D_6$

Order 108: $C_3:S_3^2$ x 7, $C_3\times S_3^2$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 81: $C_3^4$

Order 72: $\SOPlus(4,2)$ x 3, $S_3\times D_6$ x 2, $C_6:D_6$, $C_2\times C_3^2:C_4$, $D_4\times C_3^2$, $C_6^2:C_2$, $C_3:D_{12}$, $C_{12}:S_3$

Order 54: $C_3^2:C_6$ x 5, $S_3\times C_3^2$ x 5, $C_3^2:S_3$ x 4, $C_3^2\times C_6$

Order 48: $S_3\times D_4$

Order 36: $S_3^2$ x 8, $C_6:S_3$ x 7, $C_6\times S_3$ x 3, $C_3^2:C_4$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$

Order 27: $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 18: $C_3:S_3$ x 11, $C_3\times S_3$ x 8, $C_3\times C_6$ x 5

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 11, $C_2\times C_6$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 6

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

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

Order 4: $C_2^2$ x 5, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $(C_3\times S_3^2):D_6$ ($C_1$)

Order 648: $C_3:S_3^3$ ($C_2$) x 2, $C_3^4:D_4$ ($C_2$) x 2, $C_3^3:D_{12}$ ($C_2$), $C_3^4:D_4$ ($C_2$), $C_3^3:(C_4\times S_3)$ ($C_2$)

Order 324: $C_3^2:S_3^2$ ($C_2^2$) x 2, $C_3^2\times S_3^2$ ($C_2^2$) x 2, $C_3^2:S_3^2$ ($C_2^2$), $C_3^4:C_4$ ($C_2^2$), $C_3^3:C_{12}$ ($C_2^2$)

Order 216: $S_3^2:C_6$ ($S_3$) x 4

Order 162: $C_3^3:S_3$ ($D_4$), $C_3^2\wr C_2$ ($C_2^3$), $C_3^2\wr C_2$ ($D_4$)

Order 108: $C_3\times S_3^2$ ($D_6$) x 8, $C_3^2:C_{12}$ ($D_6$) x 4

Order 81: $C_3^4$ ($C_2\times D_4$)

Order 72: $\SOPlus(4,2)$ ($C_3:S_3$)

Order 54: $C_3^2:C_6$ ($C_2\times D_6$) x 4

Order 36: $S_3^2$ ($C_6:S_3$) x 2, $C_3^2:C_4$ ($C_6:S_3$)

Order 27: $C_3^3$ ($S_3\times D_4$) x 4

Order 18: $C_3:S_3$ ($C_6:D_6$), $C_3:S_3$ ($\SOPlus(4,2)$)

Order 9: $C_3^2$ ($S_3^2:C_2^2$), $C_3^2$ ($C_{12}:D_6$)

Order 3: $C_3$ ($S_3^3:C_2$) x 4

Order 1: $C_1$ ($(C_3\times S_3^2):D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $(C_3\times S_3^2):D_6$ ($C_1$)

Order 648: $C_3:S_3^3$ ($C_2$), $C_3^3:D_{12}$ ($C_2$), $C_3^4:D_4$ ($C_2$), $C_3^4:D_4$ ($C_2$), $C_3^3:(C_4\times S_3)$ ($C_2$)

Order 324: $C_3^2:S_3^2$ ($C_2^2$), $C_3^2:S_3^2$ ($C_2^2$), $C_3^2\times S_3^2$ ($C_2^2$), $C_3^4:C_4$ ($C_2^2$), $C_3^3:C_{12}$ ($C_2^2$)

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

Order 162: $C_3^3:S_3$ ($D_4$), $C_3^2\wr C_2$ ($C_2^3$), $C_3^2\wr C_2$ ($D_4$)

Order 108: $C_3\times S_3^2$ ($D_6$), $C_3^2:C_{12}$ ($D_6$)

Order 81: $C_3^4$ ($C_2\times D_4$)

Order 72: $\SOPlus(4,2)$ ($C_3:S_3$)

Order 54: $C_3^2:C_6$ ($C_2\times D_6$)

Order 36: $C_3^2:C_4$ ($C_6:S_3$), $S_3^2$ ($C_6:S_3$)

Order 27: $C_3^3$ ($S_3\times D_4$)

Order 18: $C_3:S_3$ ($C_6:D_6$), $C_3:S_3$ ($\SOPlus(4,2)$)

Order 9: $C_3^2$ ($S_3^2:C_2^2$), $C_3^2$ ($C_{12}:D_6$)

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

Order 1: $C_1$ ($(C_3\times S_3^2):D_6$)

Series

Derived series

$(C_3\times S_3^2):D_6$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$(C_3\times S_3^2):D_6$

$\rhd$

$C_3^4:D_4$

$\rhd$

$C_3^2\times S_3^2$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$(C_3\times S_3^2):D_6$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 19 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $54 \times 54$ rational character table. Alternatively, you may search for characters of this group with desired properties.