Abstract group 108.28: $C_3^2:D_6$

• Label: 108.28
• Order: \( 2^{2} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $11$
• Trans deg.: $18$
• Rank: $3$

Group information

Description:	$C_3^2:D_6$
Order:	\(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_2\times C_3^2:\GL(2,3)$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Composition factors:	$C_2$ x 2, $C_3$ x 3
Derived length:	$3$

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

Group statistics

Order	1	2	3	6
Elements	1	19	26	62	108
Conjugacy classes	1	3	6	10	20
Divisions	1	3	5	7	16
Autjugacy classes	1	2	2	3	8

Dimension	1	2	3	6
Irr. complex chars.	4	8	8	0	20
Irr. rational chars.	4	8	0	4	16

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$18$
Rank:	$3$
Inequivalent generating triples:	$567$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	6
Arbitrary	3	6	6

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{3}=c^{3}=d^{6}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{2}, d^{a}=d^{5}, d^{b}=c^{2}d \rangle$
Permutation group:	Degree $11$ $\langle(2,4)(5,9)(6,8)(10,11), (10,11), (1,2,4)(3,6,8)(5,9,7), (2,5,6)(4,8,9), (1,3,7)(2,6,5)(4,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 1 & 2 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{3})$
Transitive group:	18T52	36T79			more information
Direct product:	$C_2$ $\, \times\, $ $(C_3^2:S_3)$
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $D_6$ (4)	$\He_3$ $\,\rtimes\,$ $C_2^2$	$(C_3\times C_6)$ $\,\rtimes\,$ $S_3$ (4)	$(C_2\times \He_3)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_3:S_3)$	$C_3$ . $(C_6:S_3)$			more information
Aut. group:	$\Aut(C_3\times C_9)$	$\Aut(C_3\times C_{18})$	$\Aut(C_9:C_{12})$

Elements of the group are displayed as matrices in $\GL_{3}(\F_{3})$.

Homology

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

Subgroups

There are 167 subgroups in 55 conjugacy classes, 17 normal (7 characteristic).

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

Special subgroups

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

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

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

Classes of subgroups up to conjugation

Order 108: $C_3^2:D_6$

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

Order 36: $C_6\times S_3$ x 4

Order 27: $\He_3$

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 4

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

Order 9: $C_3^2$ x 4

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 108: $C_3^2:D_6$

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

Order 36: $C_6\times S_3$

Order 27: $\He_3$

Order 18: $C_3\times C_6$, $C_3\times S_3$

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

Order 9: $C_3^2$

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

Order 4: $C_2^2$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 108: $C_3^2:D_6$ ($C_1$)

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

Order 27: $\He_3$ ($C_2^2$)

Order 18: $C_3\times C_6$ ($S_3$) x 4

Order 9: $C_3^2$ ($D_6$) x 4

Order 6: $C_6$ ($C_3:S_3$)

Order 3: $C_3$ ($C_6:S_3$)

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

Order 1: $C_1$ ($C_3^2:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 108: $C_3^2:D_6$ ($C_1$)

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

Order 27: $\He_3$ ($C_2^2$)

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

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

Order 6: $C_6$ ($C_3:S_3$)

Order 3: $C_3$ ($C_6:S_3$)

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

Order 1: $C_1$ ($C_3^2:D_6$)

Series

Derived series

$C_3^2:D_6$

$\rhd$

$\He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3^2:D_6$

$\rhd$

$C_3^2:S_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:D_6$

$\rhd$

$\He_3$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 39 larger groups in the database.

This group is a maximal quotient of 41 larger groups in the database.

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	3B	3C	3D	3E	6A	6B	6C	6D	6E	6F	6G
	Size	1	1	9	9	2	6	6	6	6	2	6	6	6	6	18	18
	2 P	1A	1A	1A	1A	3A	3B	3C	3D	3E	3A	3B	3C	3D	3E	3A	3A
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	2A	2A	2A	2A	2A	2B	2C
108.28.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.28.1b		$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$
108.28.1c		$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
108.28.1d		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
108.28.2a		$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$0$	$0$
108.28.2b		$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$2$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$0$	$0$
108.28.2c		$2$	$2$	$0$	$0$	$2$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$2$	$-1$	$-1$	$0$	$0$
108.28.2d		$2$	$2$	$0$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$
108.28.2e		$2$	$-2$	$0$	$0$	$2$	$-1$	$-1$	$-1$	$2$	$-2$	$1$	$1$	$1$	$-2$	$0$	$0$
108.28.2f		$2$	$-2$	$0$	$0$	$2$	$-1$	$-1$	$2$	$-1$	$-2$	$1$	$1$	$-2$	$1$	$0$	$0$
108.28.2g		$2$	$-2$	$0$	$0$	$2$	$-1$	$2$	$-1$	$-1$	$-2$	$1$	$-2$	$1$	$1$	$0$	$0$
108.28.2h		$2$	$-2$	$0$	$0$	$2$	$2$	$-1$	$-1$	$-1$	$-2$	$-2$	$1$	$1$	$1$	$0$	$0$
108.28.3a		$6$	$6$	$2$	$2$	$-3$	$0$	$0$	$0$	$0$	$-3$	$0$	$0$	$0$	$0$	$-1$	$-1$
108.28.3b		$6$	$6$	$-2$	$-2$	$-3$	$0$	$0$	$0$	$0$	$-3$	$0$	$0$	$0$	$0$	$1$	$1$
108.28.3c		$6$	$-6$	$-2$	$2$	$-3$	$0$	$0$	$0$	$0$	$3$	$0$	$0$	$0$	$0$	$1$	$-1$
108.28.3d		$6$	$-6$	$2$	$-2$	$-3$	$0$	$0$	$0$	$0$	$3$	$0$	$0$	$0$	$0$	$-1$	$1$