Abstract group 1296.2919: $C_6^2:S_3^2$

• Label: 1296.2919
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^2:S_3^2$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2:D_6\times S_4$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	99	242	60	630	264	1296
Conjugacy classes	1	5	13	2	21	8	50
Divisions	1	5	9	2	13	5	35
Autjugacy classes	1	5	9	2	13	5	35

Dimension	1	2	3	4	6	8	12	18
Irr. complex chars.	12	12	12	3	8	0	1	2	50
Irr. rational chars.	4	8	4	5	8	1	3	2	35

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	18	18
Arbitrary	9	9	9

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{6}=c^{6}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{5}, d^{a}=cd, c^{b}=c^{2}d^{3}, d^{b}=c^{3}d^{5} \rangle$
Permutation group:	Degree $13$ $\langle(2,4)(3,6)(5,7)(8,9), (11,13), (2,5,9)(4,7,8), (11,12,13), (1,2,4)(3,5,8)(6,9,7), (1,3,6)(2,5,9)(4,8,7), (10,11)(12,13), (10,12)(11,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 8 & 9 \\ 9 & 8 \end{array}\right), \left(\begin{array}{rr} 7 & 29 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 17 & 32 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 27 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Transitive group:	36T2272	36T2285			more information
Direct product:	$S_4$ $\, \times\, $ $(C_3^2:C_6)$
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $S_3^2$	$(C_3^2\times S_4)$ $\,\rtimes\,$ $S_3$	$C_3^2$ $\,\rtimes\,$ $(S_3\times S_4)$	$(C_3^2\times S_4)$ $\,\rtimes\,$ $C_6$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $(C_6^2:D_6)$	$(C_3\times S_4)$ . $(C_3\times S_3)$	$(C_3\times A_4)$ . $(C_6\times S_3)$	$(C_2\times C_6)$ . $(C_3\times S_3^2)$	more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 4010 subgroups in 297 conjugacy classes, 30 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_6^2:S_3^2$
Commutator:	$G' \simeq$ $C_3^2\times A_4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_6^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times \He_3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^2:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3\times S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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

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Normal subgroups

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: $A_4\times C_3^2:C_6$, $\He_3:S_4$, $S_4\times \He_3$

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

Order 324: $A_4\times \He_3$, $C_3^2:S_3^2$

Order 216: $C_3^2\times S_4$ x 3, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:C_6$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$

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

Order 144: $S_3\times S_4$ x 2, $C_6\times S_4$, $C_{12}:D_6$, $C_{12}:D_6$

Order 108: $C_3^2:D_6$ x 5, $C_3^2\times A_4$ x 3, $C_3^2:A_4$ x 3, $C_2^2\times \He_3$ x 2, $C_3^2:C_{12}$, $C_3:S_3^2$, $C_3\times S_3^2$, $C_4\times \He_3$

Order 81: $C_3\times \He_3$

Order 72: $C_3\times S_4$ x 6, $C_3:S_4$ x 4, $S_3\times A_4$ x 3, $D_4\times C_3^2$ x 3, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_6\wr C_2$ x 2, $C_6\times A_4$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 54: $C_3^2:C_6$ x 5, $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 2, $C_2\times \He_3$ x 2, $C_3^2:S_3$

Order 48: $S_3\times D_4$ x 2, $C_2\times S_4$, $C_6\times D_4$

Order 36: $C_3\times A_4$ x 10, $C_6^2$ x 6, $C_6\times S_3$ x 6, $C_6:S_3$ x 5, $C_3\times C_{12}$ x 3, $S_3^2$ x 2, $C_3^2:C_4$, $C_3:C_{12}$

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

Order 24: $C_3\times D_4$ x 7, $C_3:D_4$ x 4, $C_2\times D_6$ x 4, $S_4$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 7, $C_3:S_3$ x 7

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 13, $D_6$ x 11, $A_4$ x 5, $C_{12}$ x 5, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 13

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

Order 6: $C_6$ x 13, $S_3$ x 10

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 648: $A_4\times C_3^2:C_6$, $\He_3:S_4$, $S_4\times \He_3$

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

Order 324: $A_4\times \He_3$, $C_3^2:S_3^2$

Order 216: $C_3^2\times S_4$ x 3, $C_6^2:C_6$ x 2, $C_6^2:C_6$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:C_6$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$

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

Order 144: $S_3\times S_4$ x 2, $C_6\times S_4$, $C_{12}:D_6$, $C_{12}:D_6$

Order 108: $C_3^2:D_6$ x 5, $C_3^2\times A_4$ x 3, $C_3^2:A_4$ x 3, $C_2^2\times \He_3$ x 2, $C_3^2:C_{12}$, $C_3:S_3^2$, $C_3\times S_3^2$, $C_4\times \He_3$

Order 81: $C_3\times \He_3$

Order 72: $C_3\times S_4$ x 6, $C_3:S_4$ x 4, $S_3\times A_4$ x 3, $D_4\times C_3^2$ x 3, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_6\wr C_2$ x 2, $C_6\times A_4$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3\times D_{12}$, $S_3\times C_{12}$

Order 54: $C_3^2:C_6$ x 5, $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 2, $C_2\times \He_3$ x 2, $C_3^2:S_3$

Order 48: $S_3\times D_4$ x 2, $C_2\times S_4$, $C_6\times D_4$

Order 36: $C_3\times A_4$ x 10, $C_6^2$ x 6, $C_6\times S_3$ x 6, $C_6:S_3$ x 5, $C_3\times C_{12}$ x 3, $S_3^2$ x 2, $C_3^2:C_4$, $C_3:C_{12}$

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

Order 24: $C_3\times D_4$ x 7, $C_3:D_4$ x 4, $C_2\times D_6$ x 4, $S_4$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 12, $C_3\times C_6$ x 7, $C_3:S_3$ x 7

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 13, $D_6$ x 11, $A_4$ x 5, $C_{12}$ x 5, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 13

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

Order 6: $C_6$ x 13, $S_3$ x 10

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 648: $A_4\times C_3^2:C_6$ ($C_2$), $\He_3:S_4$ ($C_2$), $S_4\times \He_3$ ($C_2$)

Order 432: $C_6^2:D_6$ ($C_3$)

Order 324: $A_4\times \He_3$ ($C_2^2$)

Order 216: $C_6^2:C_6$ ($C_6$), $C_3^2:S_4$ ($C_6$), $C_3^2\times S_4$ ($C_6$), $C_3^2\times S_4$ ($S_3$), $C_6^2:C_6$ ($S_3$)

Order 108: $C_3^2\times A_4$ ($C_2\times C_6$), $C_3^2\times A_4$ ($D_6$), $C_2^2\times \He_3$ ($D_6$)

Order 72: $C_6:D_6$ ($C_3\times S_3$), $C_3\times S_4$ ($C_3\times S_3$)

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

Order 36: $C_6^2$ ($C_6\times S_3$), $C_6^2$ ($S_3^2$), $C_3\times A_4$ ($C_6\times S_3$)

Order 27: $\He_3$ ($C_2\times S_4$)

Order 24: $S_4$ ($C_3^2:C_6$)

Order 18: $C_3:S_3$ ($C_3\times S_4$)

Order 12: $C_2\times C_6$ ($C_3\times S_3^2$), $A_4$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6\times S_4$), $C_3^2$ ($S_3\times S_4$)

Order 4: $C_2^2$ ($C_3^2:S_3^2$)

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

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: $A_4\times C_3^2:C_6$ ($C_2$), $\He_3:S_4$ ($C_2$), $S_4\times \He_3$ ($C_2$)

Order 432: $C_6^2:D_6$ ($C_3$)

Order 324: $A_4\times \He_3$ ($C_2^2$)

Order 216: $C_6^2:C_6$ ($C_6$), $C_3^2:S_4$ ($C_6$), $C_3^2\times S_4$ ($C_6$), $C_3^2\times S_4$ ($S_3$), $C_6^2:C_6$ ($S_3$)

Order 108: $C_3^2\times A_4$ ($C_2\times C_6$), $C_3^2\times A_4$ ($D_6$), $C_2^2\times \He_3$ ($D_6$)

Order 72: $C_6:D_6$ ($C_3\times S_3$), $C_3\times S_4$ ($C_3\times S_3$)

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

Order 36: $C_6^2$ ($C_6\times S_3$), $C_6^2$ ($S_3^2$), $C_3\times A_4$ ($C_6\times S_3$)

Order 27: $\He_3$ ($C_2\times S_4$)

Order 24: $S_4$ ($C_3^2:C_6$)

Order 18: $C_3:S_3$ ($C_3\times S_4$)

Order 12: $C_2\times C_6$ ($C_3\times S_3^2$), $A_4$ ($C_3^2:D_6$)

Order 9: $C_3^2$ ($C_6\times S_4$), $C_3^2$ ($S_3\times S_4$)

Order 4: $C_2^2$ ($C_3^2:S_3^2$)

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

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

Series

Derived series

$C_6^2:S_3^2$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_6^2:S_3^2$

$\rhd$

$S_4\times \He_3$

$\rhd$

$A_4\times \He_3$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6^2:S_3^2$

$\rhd$

$C_3^2\times A_4$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $50 \times 50$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.