Abstract group 1296.1860: $C_6^2.S_3^2$

• Label: 1296.1860
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{2} \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_6^2.S_3^2$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6^2.(C_3^3\times A_4).C_2^3$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	255	44	852	36	108	1296
Conjugacy classes	1	15	6	42	1	3	68
Divisions	1	15	6	42	1	3	68
Autjugacy classes	1	4	4	8	1	1	19

Dimension	1	2	4	6
Irr. complex chars.	16	16	4	32	68
Irr. rational chars.	16	16	4	32	68

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	$8624070$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{3}=d^{6}=e^{6}=[a,c]=[a,d]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(2,4)(5,7)(8,9)(10,11)(12,13), (3,6)(5,8)(7,9)(10,12)(11,13), (10,12)(11,13) \!\cdots\! \rangle$
Transitive group:	36T2072				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3^3:D_6)$
Semidirect product:	$(C_6^2:S_3)$ $\,\rtimes\,$ $S_3$	$(C_6^2:S_3)$ $\,\rtimes\,$ $S_3$	$(C_3^2:D_6)$ $\,\rtimes\,$ $D_6$	$(C_3^2:D_6)$ $\,\rtimes\,$ $D_6$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $S_3^2$	$C_3^2$ . $D_6^2$	$C_6$ . $(C_6.S_3^2)$	$C_3$ . $(C_6^2:D_6)$	all 6

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 14500 subgroups in 1104 conjugacy classes, 114 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_3^3:D_6$
Commutator:	$G' \simeq$ $C_3\wr C_3$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2.C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1296: $C_6^2.S_3^2$

Order 648: $C_2\times C_3^3:D_6$ x 12, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $C_2^2\times C_3\wr S_3$

Order 432: $C_3:D_6^2$, $C_6^2:D_6$

Order 324: $C_3^3:D_6$ x 16, $C_3^3:D_6$ x 6, $\He_3:D_6$ x 6, $C_3^3:D_6$ x 6, $C_3^2.C_6^2$

Order 216: $C_6:S_3^2$ x 12, $C_6.S_3^2$ x 12, $C_6^2:C_6$ x 2, $C_6^2:S_3$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6^2:S_3$, $D_{18}:C_6$

Order 162: $C_3^3:S_3$ x 4, $C_3^3:C_6$ x 4, $C_3\wr S_3$ x 4, $C_3^3:C_6$ x 3

Order 144: $D_6^2$ x 5, $C_6^2:C_2^2$

Order 108: $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 16, $C_3^2:D_6$ x 12, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 6, $C_3^2:D_6$ x 6, $C_{18}:C_6$ x 6, $C_3\times C_6^2$, $C_{18}:C_6$, $C_2^2\times \He_3$

Order 81: $C_3\wr C_3$

Order 72: $S_3\times D_6$ x 60, $C_6:D_6$ x 19, $C_6\times D_6$ x 10, $C_2\times C_6^2$, $C_2\times D_{18}$

Order 54: $C_3^2:C_6$ x 8, $C_3^2:S_3$ x 4, $C_9:C_6$ x 4, $C_3^2:S_3$ x 4, $C_3^2:C_6$ x 4, $S_3\times C_3^2$ x 4, $C_3^2\times C_6$ x 3, $C_9:C_6$ x 3, $C_2\times \He_3$ x 3

Order 48: $C_2^2\times D_6$ x 6

Order 36: $S_3^2$ x 80, $C_6\times S_3$ x 60, $C_6:S_3$ x 58, $C_6^2$ x 12, $D_{18}$ x 6, $C_2\times C_{18}$

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

Order 24: $C_2\times D_6$ x 84, $C_2^2\times C_6$ x 6

Order 18: $C_3\times S_3$ x 40, $C_3:S_3$ x 28, $C_3\times C_6$ x 22, $D_9$ x 4, $C_{18}$ x 3

Order 16: $C_2^4$

Order 12: $D_6$ x 168, $C_2\times C_6$ x 42

Order 9: $C_3^2$ x 6, $C_9$

Order 8: $C_2^3$ x 15

Order 6: $S_3$ x 48, $C_6$ x 42

Order 4: $C_2^2$ x 35

Order 3: $C_3$ x 6

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_6^2.S_3^2$

Order 648: $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $C_2^2\times C_3\wr S_3$, $C_2\times C_3^3:D_6$

Order 432: $C_3:D_6^2$, $C_6^2:D_6$

Order 324: $C_3^2.C_6^2$, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:D_6$

Order 216: $C_6^2:C_6$ x 2, $C_6^2:S_3$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6:S_3^2$, $C_6^2:S_3$, $D_{18}:C_6$, $C_6.S_3^2$

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

Order 144: $D_6^2$ x 3, $C_6^2:C_2^2$

Order 108: $C_3^2:D_6$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$, $C_{18}:C_6$, $C_2^2\times \He_3$, $C_3^2:D_6$, $C_{18}:C_6$, $C_3^2:D_6$

Order 81: $C_3\wr C_3$

Order 72: $C_6:D_6$ x 6, $C_6\times D_6$ x 6, $S_3\times D_6$ x 3, $C_2\times C_6^2$, $C_2\times D_{18}$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$, $C_9:C_6$, $C_3^2\times C_6$, $C_3^2:S_3$, $C_3^2:C_6$, $S_3\times C_3^2$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2^2\times D_6$ x 4

Order 36: $C_6:S_3$ x 6, $C_6\times S_3$ x 6, $C_6^2$ x 5, $S_3^2$ x 3, $C_2\times C_{18}$, $D_{18}$

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

Order 24: $C_2\times D_6$ x 12, $C_2^2\times C_6$ x 4

Order 18: $C_3\times S_3$ x 6, $C_3\times C_6$ x 5, $C_3:S_3$ x 5, $C_{18}$, $D_9$

Order 16: $C_2^4$

Order 12: $D_6$ x 12, $C_2\times C_6$ x 8

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

Order 8: $C_2^3$ x 4

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

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_6^2.S_3^2$ ($C_1$)

Order 648: $C_2\times C_3^3:D_6$ ($C_2$) x 12, $C_2\times C_3^3:D_6$ ($C_2$), $C_2^2\times C_3^3:C_6$ ($C_2$), $C_2^2\times C_3\wr S_3$ ($C_2$)

Order 324: $C_3^3:D_6$ ($C_2^2$) x 16, $C_3^3:D_6$ ($C_2^2$) x 6, $\He_3:D_6$ ($C_2^2$) x 6, $C_3^3:D_6$ ($C_2^2$) x 6, $C_3^2.C_6^2$ ($C_2^2$)

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

Order 162: $C_3^3:S_3$ ($C_2^3$) x 4, $C_3^3:C_6$ ($C_2^3$) x 4, $C_3\wr S_3$ ($C_2^3$) x 4, $C_3^3:C_6$ ($C_2^3$) x 3

Order 108: $C_3^2:D_6$ ($D_6$) x 6, $C_3^2:D_6$ ($D_6$) x 6, $C_3\times C_6^2$ ($D_6$), $C_2^2\times \He_3$ ($D_6$)

Order 81: $C_3\wr C_3$ ($C_2^4$)

Order 54: $C_3^2:S_3$ ($C_2\times D_6$) x 4, $C_3^2:S_3$ ($C_2\times D_6$) x 4, $C_3^2\times C_6$ ($C_2\times D_6$) x 3, $C_2\times \He_3$ ($C_2\times D_6$) x 3

Order 36: $C_6^2$ ($S_3^2$)

Order 27: $C_3^3$ ($C_2^2\times D_6$), $\He_3$ ($C_2^2\times D_6$)

Order 18: $C_3\times C_6$ ($S_3\times D_6$) x 3

Order 12: $C_2\times C_6$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($D_6^2$)

Order 6: $C_6$ ($C_6.S_3^2$) x 3

Order 4: $C_2^2$ ($C_3^3:D_6$)

Order 3: $C_3$ ($C_6^2:D_6$)

Order 2: $C_2$ ($C_2\times C_3^3:D_6$) x 3

Order 1: $C_1$ ($C_6^2.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_6^2.S_3^2$ ($C_1$)

Order 648: $C_2\times C_3^3:D_6$ ($C_2$), $C_2^2\times C_3^3:C_6$ ($C_2$), $C_2^2\times C_3\wr S_3$ ($C_2$), $C_2\times C_3^3:D_6$ ($C_2$)

Order 324: $C_3^2.C_6^2$ ($C_2^2$), $C_3^3:D_6$ ($C_2^2$), $\He_3:D_6$ ($C_2^2$), $C_3^3:D_6$ ($C_2^2$), $C_3^3:D_6$ ($C_2^2$)

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

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

Order 108: $C_3\times C_6^2$ ($D_6$), $C_3^2:D_6$ ($D_6$), $C_2^2\times \He_3$ ($D_6$), $C_3^2:D_6$ ($D_6$)

Order 81: $C_3\wr C_3$ ($C_2^4$)

Order 54: $C_3^2:S_3$ ($C_2\times D_6$), $C_3^2\times C_6$ ($C_2\times D_6$), $C_3^2:S_3$ ($C_2\times D_6$), $C_2\times \He_3$ ($C_2\times D_6$)

Order 36: $C_6^2$ ($S_3^2$)

Order 27: $C_3^3$ ($C_2^2\times D_6$), $\He_3$ ($C_2^2\times D_6$)

Order 18: $C_3\times C_6$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($D_6^2$)

Order 6: $C_6$ ($C_6.S_3^2$)

Order 4: $C_2^2$ ($C_3^3:D_6$)

Order 3: $C_3$ ($C_6^2:D_6$)

Order 2: $C_2$ ($C_2\times C_3^3:D_6$)

Order 1: $C_1$ ($C_6^2.S_3^2$)

Series

Derived series

$C_6^2.S_3^2$

$\rhd$

$C_3\wr C_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^2.S_3^2$

$\rhd$

$C_2^2\times C_3\wr S_3$

$\rhd$

$C_3^2.C_6^2$

$\rhd$

$C_3^3:C_6$

$\rhd$

$C_3\wr C_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2.S_3^2$

$\rhd$

$C_3\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 15 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 $68 \times 68$ rational character table. Alternatively, you may search for characters of this group with desired properties.