Abstract group 432.523: $C_6^2:D_6$

• Label: 432.523
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $13$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_6^2:D_6$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 3
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	75	80	36	168	72	432
Conjugacy classes	1	5	4	2	6	2	20
Divisions	1	5	4	2	6	2	20
Autjugacy classes	1	5	4	2	6	2	20

Dimension	1	2	3	4	6	12
Irr. complex chars.	4	4	4	1	6	1	20
Irr. rational chars.	4	4	4	1	6	1	20

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$54$

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:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{6}=d^{6}=[a,d]=[c,d]=1, b^{a}=b^{5}, c^{a}=c^{5}d^{3}, c^{b}=c^{2}d^{3}, d^{b}=c^{5}d^{5} \rangle$
Permutation group:	Degree $13$ $\langle(3,7)(5,6)(8,9), (2,4)(3,7)(5,8)(6,9)(11,12), (1,2,4)(3,6,8)(5,9,7)(11,12,13), (2,5,6)(4,8,9), (1,3,7)(2,6,5)(4,8,9), (10,11)(12,13), (10,12)(11,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Transitive group:	18T152	18T153	18T154	18T155	all 15
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_6^2$ $\,\rtimes\,$ $D_6$	$(C_3:S_4)$ $\,\rtimes\,$ $S_3$	$(C_3:S_3)$ $\,\rtimes\,$ $S_4$	$(C_6:D_6)$ $\,\rtimes\,$ $S_3$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $(S_3\times S_4)$	$(C_2\times C_6)$ . $S_3^2$			more information
Aut. group:	$\Aut(C_3:S_4)$	$\Aut(C_3^2:S_4)$	$\Aut(C_3^2:S_4)$

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

Homology

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

Subgroups

There are 1289 subgroups in 134 conjugacy classes, 15 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_6^2:D_6$
Commutator:	$G' \simeq$ $C_3^2:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^2:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

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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Subgroup information Click on a subgroup in the diagram to see information about it.

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Classes of subgroups up to conjugation

Order 432: $C_6^2:D_6$

Order 216: $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$

Order 144: $S_3\times S_4$, $D_6:D_6$

Order 108: $C_3^2:A_4$, $C_3^2:D_6$

Order 72: $C_3\times S_4$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6:D_6$, $S_3\times D_6$, $S_3\times A_4$, $C_3:S_4$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$

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

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

Order 27: $\He_3$

Order 24: $C_3:D_4$ x 4, $C_2\times D_6$ x 4, $S_4$ x 3, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times A_4$

Order 18: $C_3\times S_3$ x 5, $C_3:S_3$ x 3, $C_3\times C_6$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 13, $C_2\times C_6$ x 4, $A_4$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 3

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

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

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 432: $C_6^2:D_6$

Order 216: $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$

Order 144: $S_3\times S_4$, $D_6:D_6$

Order 108: $C_3^2:A_4$, $C_3^2:D_6$

Order 72: $C_3\times S_4$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $C_6:D_6$, $S_3\times D_6$, $S_3\times A_4$, $C_3:S_4$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 2, $C_3^2:S_3$

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

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

Order 27: $\He_3$

Order 24: $C_3:D_4$ x 4, $C_2\times D_6$ x 4, $S_4$ x 3, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times A_4$

Order 18: $C_3\times S_3$ x 5, $C_3:S_3$ x 3, $C_3\times C_6$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 13, $C_2\times C_6$ x 4, $A_4$ x 2, $C_{12}$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 3

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

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

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: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $C_6^2:D_6$ ($C_1$)

Order 216: $C_6^2:C_6$ ($C_2$), $C_3^2:S_4$ ($C_2$), $C_3^2:S_4$ ($C_2$)

Order 108: $C_3^2:A_4$ ($C_2^2$)

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

Order 36: $C_6^2$ ($D_6$), $C_3\times A_4$ ($D_6$)

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

Order 12: $C_2\times C_6$ ($S_3^2$)

Order 9: $C_3^2$ ($C_2\times S_4$)

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

Order 3: $C_3$ ($S_3\times S_4$)

Order 1: $C_1$ ($C_6^2:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $C_6^2:D_6$ ($C_1$)

Order 216: $C_6^2:C_6$ ($C_2$), $C_3^2:S_4$ ($C_2$), $C_3^2:S_4$ ($C_2$)

Order 108: $C_3^2:A_4$ ($C_2^2$)

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

Order 36: $C_6^2$ ($D_6$), $C_3\times A_4$ ($D_6$)

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

Order 12: $C_2\times C_6$ ($S_3^2$)

Order 9: $C_3^2$ ($C_2\times S_4$)

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

Order 3: $C_3$ ($S_3\times S_4$)

Order 1: $C_1$ ($C_6^2:D_6$)

Series

Derived series

$C_6^2:D_6$

$\rhd$

$C_3^2:A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_6^2:D_6$

$\rhd$

$C_3^2:S_4$

$\rhd$

$C_3^2:A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6^2:D_6$

$\rhd$

$C_3^2:A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 29 larger groups in the database.

This group is a maximal quotient of 53 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 $20 \times 20$ rational character table. Alternatively, you may search for characters of this group with desired properties.