Abstract group 108.25: $C_3^2:D_6$

• Label: 108.25
• Order: \( 2^{2} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $11$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_3^2:D_6$
Order:	\(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$C_6.S_3^2$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
Composition factors:	$C_2$ x 2, $C_3$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	19	26	62	108
Conjugacy classes	1	3	6	10	20
Divisions	1	3	4	6	14
Autjugacy classes	1	2	4	5	12

Dimension	1	2	4	6
Irr. complex chars.	12	6	0	2	20
Irr. rational chars.	4	6	2	2	14

Minimal presentations

Permutation degree:	$11$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$12$

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:	$\langle a, b, c \mid a^{6}=b^{3}=c^{6}=[b,c]=1, b^{a}=b^{2}c^{2}, c^{a}=c^{5} \rangle$
Permutation group:	Degree $11$ $\langle(3,7)(5,6)(8,9), (10,11), (1,2,4)(3,6,8)(5,9,7), (2,5,6)(4,8,9), (1,3,7)(2,6,5)(4,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 1 & 2 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{3})$
Transitive group:	18T41	18T42	36T71	36T73	all 5
Direct product:	$C_2$ $\, \times\, $ $(C_3^2:C_6)$
Semidirect product:	$C_3^2$ $\,\rtimes\,$ $D_6$	$\He_3$ $\,\rtimes\,$ $C_2^2$	$(C_6:S_3)$ $\,\rtimes\,$ $C_3$	$(C_3\times C_6)$ $\,\rtimes\,$ $S_3$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_3\times S_3)$	$C_3$ . $(C_6\times S_3)$			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:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 143 subgroups in 39 conjugacy classes, 16 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^2:C_6$
Commutator:	$G' \simeq$ $C_3^2$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_6\times S_3$
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:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
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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Classes of subgroups up to conjugation

Order 108: $C_3^2:D_6$

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

Order 36: $C_6:S_3$, $C_6\times S_3$

Order 27: $\He_3$

Order 18: $C_3\times C_6$ x 3, $C_3:S_3$ x 2, $C_3\times S_3$ x 2

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

Order 9: $C_3^2$ x 3

Order 6: $C_6$ x 6, $S_3$ x 4

Order 4: $C_2^2$

Order 3: $C_3$ x 4

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: $C_3^2:C_6$, $C_2\times \He_3$

Order 36: $C_6:S_3$, $C_6\times S_3$

Order 27: $\He_3$

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

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

Order 9: $C_3^2$ x 3

Order 6: $C_6$ x 5, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 4

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

Order 36: $C_6:S_3$ ($C_3$)

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

Order 18: $C_3:S_3$ ($C_6$) x 2, $C_3\times C_6$ ($C_6$), $C_3\times C_6$ ($S_3$)

Order 9: $C_3^2$ ($C_2\times C_6$), $C_3^2$ ($D_6$)

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

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

Order 2: $C_2$ ($C_3^2:C_6$)

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

Order 36: $C_6:S_3$ ($C_3$)

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

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

Order 9: $C_3^2$ ($C_2\times C_6$), $C_3^2$ ($D_6$)

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

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

Order 2: $C_2$ ($C_3^2:C_6$)

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

Series

Derived series

$C_3^2:D_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_3^2:D_6$

$\rhd$

$C_2\times \He_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^2:D_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 50 larger groups in the database.

This group is a maximal quotient of 51 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	6A	6B	6C	6D	6E	6F
	Size	1	1	9	9	2	6	6	12	2	6	6	12	18	18
	2 P	1A	1A	1A	1A	3A	3B	3C	3D	3A	3B	3C	3D	3B	3B
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	2A	2A	2A	2A	2B	2C
108.25.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.25.1b		$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$
108.25.1c		$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$-1$
108.25.1d		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
108.25.1e		$2$	$2$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$
108.25.1f		$2$	$-2$	$-2$	$2$	$2$	$-1$	$-1$	$-1$	$-2$	$1$	$1$	$1$	$1$	$-1$
108.25.1g		$2$	$-2$	$2$	$-2$	$2$	$-1$	$-1$	$-1$	$-2$	$1$	$1$	$1$	$-1$	$1$
108.25.1h		$2$	$2$	$-2$	$-2$	$2$	$-1$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$1$	$1$
108.25.2a		$2$	$2$	$0$	$0$	$2$	$2$	$-1$	$-1$	$2$	$2$	$-1$	$-1$	$0$	$0$
108.25.2b		$2$	$-2$	$0$	$0$	$2$	$2$	$-1$	$-1$	$-2$	$-2$	$1$	$1$	$0$	$0$
108.25.2c		$4$	$4$	$0$	$0$	$4$	$-2$	$1$	$1$	$4$	$-2$	$1$	$1$	$0$	$0$
108.25.2d		$4$	$-4$	$0$	$0$	$4$	$-2$	$1$	$1$	$-4$	$2$	$-1$	$-1$	$0$	$0$
108.25.6a		$6$	$6$	$0$	$0$	$-3$	$0$	$0$	$0$	$-3$	$0$	$0$	$0$	$0$	$0$
108.25.6b		$6$	$-6$	$0$	$0$	$-3$	$0$	$0$	$0$	$3$	$0$	$0$	$0$	$0$	$0$