Abstract group 216.132: $C_3^2:D_{12}$

• Label: 216.132
• Order: \( 2^{3} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $13$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_3^2:D_{12}$
Order:	\(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Composition factors:	$C_2$ x 3, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	37	26	18	98	36	216
Conjugacy classes	1	3	8	1	12	2	27
Divisions	1	3	7	1	9	1	22
Autjugacy classes	1	2	5	1	6	1	16

Dimension	1	2	4	8
Irr. complex chars.	4	13	10	0	27
Irr. rational chars.	4	7	9	2	22

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	8
Arbitrary	4	6	6

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{3}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{5}d^{3}, d^{a}=d^{5}, c^{b}=c^{2}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $13$ $\langle(3,4)(5,6)(8,10), (1,2)(3,6)(4,5)(7,8)(9,10)(12,13), (7,9)(8,10), (1,3,4)(2,5,6), (11,12,13), (1,4,3)(2,5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 2 & 1 \\ 0 & 2 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 2 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 2 & 1 \\ 2 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 1 & 2 \\ 2 & 2 & 1 & 2 \\ 1 & 2 & 0 & 1 \\ 1 & 2 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 2 & 2 \\ 2 & 1 & 1 & 0 \\ 0 & 2 & 2 & 2 \\ 1 & 1 & 2 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 2 & 8 \end{array}\right), \left(\begin{array}{rr} 1 & 3 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 6 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 8 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 7 & 3 \\ 6 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/9\Z)$
Transitive group:	24T550	36T240			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $D_4$	$C_3^2$ $\,\rtimes\,$ $D_{12}$	$(C_6:S_3)$ $\,\rtimes\,$ $S_3$ (2)	$C_3$ $\,\rtimes\,$ $(D_6:S_3)$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $S_3^2$ (3)	$(C_3\times C_6)$ . $D_6$ (3)	$C_2$ . $(C_3:S_3^2)$	$(C_3^2\times C_6)$ . $C_2^2$	more information

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{9}\Z)$.

Homology

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

Subgroups

There are 428 subgroups in 94 conjugacy classes, 23 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3:S_3^2$
Commutator:	$G' \simeq$ $C_3^2\times C_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_3:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^2:D_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 216: $C_3^2:D_{12}$

Order 108: $C_3^2:D_6$ x 2, $C_3^2:C_{12}$

Order 72: $C_3:D_{12}$ x 2, $D_6:S_3$

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

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

Order 27: $C_3^3$

Order 24: $C_3:D_4$ x 2, $D_{12}$

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 7, $C_3:S_3$ x 2

Order 12: $D_6$ x 6, $C_3:C_4$ x 3, $C_2\times C_6$ x 2, $C_{12}$

Order 9: $C_3^2$ x 7

Order 8: $D_4$

Order 6: $C_6$ x 9, $S_3$ x 8

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 216: $C_3^2:D_{12}$

Order 108: $C_3^2:D_6$, $C_3^2:C_{12}$

Order 72: $C_3:D_{12}$, $D_6:S_3$

Order 54: $C_3^2\times C_6$, $C_3^2:C_6$

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

Order 27: $C_3^3$

Order 24: $C_3:D_4$, $D_{12}$

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

Order 12: $D_6$ x 3, $C_3:C_4$ x 2, $C_2\times C_6$, $C_{12}$

Order 9: $C_3^2$ x 5

Order 8: $D_4$

Order 6: $C_6$ x 6, $S_3$ x 3

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 216: $C_3^2:D_{12}$ ($C_1$)

Order 108: $C_3^2:D_6$ ($C_2$) x 2, $C_3^2:C_{12}$ ($C_2$)

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

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

Order 27: $C_3^3$ ($D_4$)

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

Order 9: $C_3^2$ ($C_3:D_4$) x 2, $C_3^2$ ($D_{12}$)

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

Order 3: $C_3$ ($C_3:D_{12}$) x 2, $C_3$ ($D_6:S_3$)

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

Order 1: $C_1$ ($C_3^2:D_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 216: $C_3^2:D_{12}$ ($C_1$)

Order 108: $C_3^2:D_6$ ($C_2$), $C_3^2:C_{12}$ ($C_2$)

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

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

Order 27: $C_3^3$ ($D_4$)

Order 18: $C_3\times C_6$ ($D_6$) x 2

Order 9: $C_3^2$ ($C_3:D_4$), $C_3^2$ ($D_{12}$)

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

Order 3: $C_3$ ($C_3:D_{12}$), $C_3$ ($D_6:S_3$)

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

Order 1: $C_1$ ($C_3^2:D_{12}$)

Series

Derived series

$C_3^2:D_{12}$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_1$

Chief series

$C_3^2:D_{12}$

$\rhd$

$C_3^2:D_6$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:D_{12}$

$\rhd$

$C_3^2\times C_6$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 44 larger groups in the database.

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

Character theory

Complex character table

See the $27 \times 27$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.