Abstract group $C_3:D_{12}$

• Label: 72.23
• Order: \( 2^{3} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{4} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $10$
• Trans deg.: $12$
• Rank: $2$

Group information

Description:	$C_3:D_{12}$
Order:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6^2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) (generators)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2
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	25	8	6	20	12	72
Conjugacy classes	1	3	3	1	5	2	15
Divisions	1	3	3	1	4	1	13
Autjugacy classes	1	3	3	1	4	1	13

Dimension	1	2	4
Irr. complex chars.	4	9	2	15
Irr. rational chars.	4	5	4	13

Minimal Presentations

Permutation degree:	$10$
Transitive degree:	$12$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{12}=c^{3}=[a,c]=1, b^{a}=b^{11}, c^{b}=c^{2} \rangle$
Permutation group:	Degree $10$ $\langle(1,2)(3,4)(9,10), (1,3,4,2)(6,7), (1,4)(2,3), (5,6,7), (8,9,10)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 2 & 2 \\ 2 & 0 & 1 & 2 \\ 0 & 1 & 2 & 1 \\ 1 & 2 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 2 \\ 1 & 0 & 1 & 1 \\ 2 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 2 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 1 & 2 \\ 2 & 2 & 1 & 2 \\ 1 & 2 & 0 & 1 \\ 1 & 2 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	12T38	24T74	36T33	36T38	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_6$ $\,\rtimes\,$ $S_3$	$C_3$ $\,\rtimes\,$ $D_{12}$	$C_3^2$ $\,\rtimes\,$ $D_4$	$(C_6\times S_3)$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $D_6$ (2)	$C_2$ . $S_3^2$	$(C_3\times C_6)$ . $C_2^2$		more information

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

Homology

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

Subgroups

There are 126 subgroups in 37 conjugacy classes, 14 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

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 72: $C_3:D_{12}$

Order 36: $C_3:C_{12}$, $C_6:S_3$, $C_6\times S_3$

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

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

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

Order 9: $C_3^2$

Order 8: $D_4$

Order 6: $S_3$ x 5, $C_6$ x 4

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 72: $C_3:D_{12}$

Order 36: $C_3:C_{12}$, $C_6:S_3$, $C_6\times S_3$

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

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

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

Order 9: $C_3^2$

Order 8: $D_4$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 72: $C_3:D_{12}$ ($C_1$)

Order 36: $C_3:C_{12}$ ($C_2$), $C_6:S_3$ ($C_2$), $C_6\times S_3$ ($C_2$)

Order 18: $C_3\times C_6$ ($C_2^2$)

Order 12: $D_6$ ($S_3$), $C_3:C_4$ ($S_3$)

Order 9: $C_3^2$ ($D_4$)

Order 6: $C_6$ ($D_6$) x 2

Order 3: $C_3$ ($C_3:D_4$), $C_3$ ($D_{12}$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 72: $C_3:D_{12}$ ($C_1$)

Order 36: $C_3:C_{12}$ ($C_2$), $C_6:S_3$ ($C_2$), $C_6\times S_3$ ($C_2$)

Order 18: $C_3\times C_6$ ($C_2^2$)

Order 12: $D_6$ ($S_3$), $C_3:C_4$ ($S_3$)

Order 9: $C_3^2$ ($D_4$)

Order 6: $C_6$ ($D_6$) x 2

Order 3: $C_3$ ($C_3:D_4$), $C_3$ ($D_{12}$)

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

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

Series

Derived series	$C_3:D_{12}$	$\rhd$	$C_3\times C_6$	$\rhd$	$C_1$
Chief series	$C_3:D_{12}$	$\rhd$	$C_6\times S_3$	$\rhd$	$C_3\times C_6$	$\rhd$	$C_3^2$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3:D_{12}$	$\rhd$	$C_3\times C_6$	$\rhd$	$C_3^2$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 64 larger groups in the database.

This group is a maximal quotient of 66 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	3A	3B	3C	4A	6A	6B	6C	6D1	6D-1	12A1	12A5
	Size	1	1	6	18	2	2	4	6	2	2	4	6	6	6	6
	2 P	1A	1A	1A	1A	3A	3B	3C	2A	3B	3A	3C	3A	3A	6A	6A
	3 P	1A	2A	2B	2C	1A	1A	1A	4A	2A	2A	2A	2B	2B	4A	4A
	Type
72.23.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.23.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$
72.23.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
72.23.1d	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.23.2a	R	$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$2$	$-1$	$2$	$-1$	$0$	$0$	$-1$	$-1$
72.23.2b	R	$2$	$2$	$2$	$0$	$-1$	$2$	$-1$	$0$	$2$	$-1$	$-1$	$-1$	$-1$	$0$	$0$
72.23.2c	R	$2$	$-2$	$0$	$0$	$2$	$2$	$2$	$0$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$
72.23.2d	R	$2$	$2$	$-2$	$0$	$-1$	$2$	$-1$	$0$	$2$	$-1$	$-1$	$1$	$1$	$0$	$0$
72.23.2e	R	$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$-2$	$-1$	$2$	$-1$	$0$	$0$	$1$	$1$
72.23.2f1	R	$2$	$-2$	$0$	$0$	$2$	$-1$	$-1$	$0$	$1$	$-2$	$1$	$0$	$0$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$
72.23.2f2	R	$2$	$-2$	$0$	$0$	$2$	$-1$	$-1$	$0$	$1$	$-2$	$1$	$0$	$0$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$
72.23.2g1	C	$2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$0$	$-2$	$1$	$1$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$
72.23.2g2	C	$2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$0$	$-2$	$1$	$1$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$
72.23.4a	R	$4$	$4$	$0$	$0$	$-2$	$-2$	$1$	$0$	$-2$	$-2$	$1$	$0$	$0$	$0$	$0$
72.23.4b	R	$4$	$-4$	$0$	$0$	$-2$	$-2$	$1$	$0$	$2$	$2$	$-1$	$0$	$0$	$0$	$0$

Rational character table

		1A	2A	2B	2C	3A	3B	3C	4A	6A	6B	6C	6D	12A
	Size	1	1	6	18	2	2	4	6	2	2	4	12	12
	2 P	1A	1A	1A	1A	3A	3B	3C	2A	3B	3A	3C	3A	6A
	3 P	1A	2A	2B	2C	1A	1A	1A	4A	2A	2A	2A	2B	4A
72.23.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.23.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$
72.23.1c		$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
72.23.1d		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$
72.23.2a		$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$2$	$-1$	$2$	$-1$	$0$	$-1$
72.23.2b		$2$	$2$	$2$	$0$	$-1$	$2$	$-1$	$0$	$2$	$-1$	$-1$	$-1$	$0$
72.23.2c		$2$	$-2$	$0$	$0$	$2$	$2$	$2$	$0$	$-2$	$-2$	$-2$	$0$	$0$
72.23.2d		$2$	$2$	$-2$	$0$	$-1$	$2$	$-1$	$0$	$2$	$-1$	$-1$	$1$	$0$
72.23.2e		$2$	$2$	$0$	$0$	$2$	$-1$	$-1$	$-2$	$-1$	$2$	$-1$	$0$	$1$
72.23.2f		$4$	$-4$	$0$	$0$	$4$	$-2$	$-2$	$0$	$2$	$-4$	$2$	$0$	$0$
72.23.2g		$4$	$-4$	$0$	$0$	$-2$	$4$	$-2$	$0$	$-4$	$2$	$2$	$0$	$0$
72.23.4a		$4$	$4$	$0$	$0$	$-2$	$-2$	$1$	$0$	$-2$	$-2$	$1$	$0$	$0$
72.23.4b		$4$	$-4$	$0$	$0$	$-2$	$-2$	$1$	$0$	$2$	$2$	$-1$	$0$	$0$