Abstract group $C_3:C_4$

• Label: 12.1
• Order: \( 2^{2} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $7$
• Trans deg.: $12$
• Rank: $2$

Group information

Description:	$C_3:C_4$
Order:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) (generators)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$C_2$ x 2, $C_3$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.

Group statistics

Order	1	2	3	4	6
Elements	1	1	2	6	2	12
Conjugacy classes	1	1	1	2	1	6
Divisions	1	1	1	1	1	5
Autjugacy classes	1	1	1	1	1	5

Dimension	1	2
Irr. complex chars.	4	2	6
Irr. rational chars.	2	3	5

Minimal Presentations

Permutation degree:	$7$
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	2	4	4
Arbitrary	2	4	4

Constructions

Groups of Lie type:	$\CSOPlus(2,7)$, $\CSOMinus(2,5)$
Presentation:	$\langle a, b \mid a^{4}=b^{3}=1, b^{a}=b^{2} \rangle$
Permutation group:	$\langle(2,3)(4,5,6,7), (4,6)(5,7), (1,2,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & -1 & 0 \\ 0 & 0 & -1 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} -1 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rr} 4 & 3 \\ 1 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 1 \\ 3 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{5})$
Transitive group:	12T5				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3$ $\,\rtimes\,$ $C_4$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $S_3$	$C_6$ . $C_2$			more information

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

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 8 subgroups in 6 conjugacy classes, 5 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_6$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3:C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 12: $C_3:C_4$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 12: $C_3:C_4$

Order 6: $C_6$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 12: $C_3:C_4$ ($C_1$)

Order 6: $C_6$ ($C_2$)

Order 3: $C_3$ ($C_4$)

Order 2: $C_2$ ($S_3$)

Order 1: $C_1$ ($C_3:C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 12: $C_3:C_4$ ($C_1$)

Order 6: $C_6$ ($C_2$)

Order 3: $C_3$ ($C_4$)

Order 2: $C_2$ ($S_3$)

Order 1: $C_1$ ($C_3:C_4$)

Series

Derived series	$C_3:C_4$	$\rhd$	$C_3$	$\rhd$	$C_1$
Chief series	$C_3:C_4$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3:C_4$	$\rhd$	$C_3$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 115 larger groups in the database.

This group is a maximal quotient of 129 larger groups in the database.

Character theory

Complex character table

		1A	2A	3A	4A1	4A-1	6A
	Size	1	1	2	3	3	2
	2 P	1A	1A	3A	2A	2A	3A
	3 P	1A	2A	1A	4A-1	4A1	2A
	Type
12.1.1a	R	$1$	$1$	$1$	$1$	$1$	$1$
12.1.1b	R	$1$	$1$	$1$	$-1$	$-1$	$1$
12.1.1c1	C	$1$	$-1$	$1$	$-i$	$i$	$-1$
12.1.1c2	C	$1$	$-1$	$1$	$i$	$-i$	$-1$
12.1.2a	R	$2$	$2$	$-1$	$0$	$0$	$-1$
12.1.2b	S	$2$	$-2$	$-1$	$0$	$0$	$1$

Rational character table

		1A	2A	3A	4A	6A
	Size	1	1	2	6	2
	2 P	1A	1A	3A	2A	3A
	3 P	1A	2A	1A	4A	2A
	Schur
12.1.1a		$1$	$1$	$1$	$1$	$1$
12.1.1b		$1$	$1$	$1$	$-1$	$1$
12.1.1c		$2$	$-2$	$2$	$0$	$-2$
12.1.2a		$2$	$2$	$-1$	$0$	$-1$
12.1.2b	2	$2$	$-2$	$-1$	$0$	$1$

Additional information

This group is sometimes referred to as $\widetilde{S}_3$ since it is the smallest nonsplit central extension of $S_3$. It is embeddable in $\GL_2(k)$ for any field of characteristic larger than $3$ (Serre, Finite Groups: An Introduction, exercise 10.3.2). The character 12.1.2b is an example of a character with Schur index larger than 1.