Abstract group $C_2\times S_4$

• Label: 48.48
• Order: \( 2^{4} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{4} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $6$
• Trans deg.: $6$
• Rank: $2$

Group information

Description:	$C_2\times S_4$
Order:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) (generators)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$C_2$ x 4, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	19	8	12	8	48
Conjugacy classes	1	5	1	2	1	10
Divisions	1	5	1	2	1	10
Autjugacy classes	1	4	1	1	1	8

Dimension	1	2	3
Irr. complex chars.	4	2	4	10
Irr. rational chars.	4	2	4	10

Minimal Presentations

Permutation degree:	$6$
Transitive degree:	$6$
Rank:	$2$
Inequivalent generating pairs:	$9$

Minimal degrees of faithful linear representations

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

Constructions

Groups of Lie type:	$\GO(3,3)$, $\CO(3,3)$
Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{2}=d^{2}=[c,d]=1, b^{a}=b^{5}, c^{a}=d, d^{a}=c, c^{b}=d, d^{b}=cd \rangle$
Permutation group:	$\langle(1,2)(3,6)(4,5), (1,2)(3,4)(5,6), (1,5,3)(2,6,4), (1,2)(3,4), (1,2)(5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrr} -1 & 1 & -1 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrr} 1 & -1 & 1 \\ 0 & -1 & 0 \\ -1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{3}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{array}\right), \left(\begin{array}{rrr} 1 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 2 & 2 \end{array}\right), \left(\begin{array}{rrr} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 1 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 2 \\ 2 & 2 & 2 \\ 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{3})$
Transitive group:	6T11	8T24	12T21	12T22	all 10
Direct product:	$C_2$ $\, \times\, $ $S_4$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $S_3$	$A_4$ $\,\rtimes\,$ $C_2^2$	$C_2^2$ $\,\rtimes\,$ $D_6$	$(C_2\times A_4)$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	$C_2$$\ \wr\ $$S_3$
Aut. group:	$\Aut(D_4:C_2)$	$\Aut(C_3\times Q_8)$	$\Aut(C_2.S_4)$	$\Aut(\GL(2,3))$	all 9

Elements of the group are displayed as permutations of degree 6.

Homology

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

Subgroups

There are 98 subgroups in 33 conjugacy classes, 9 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 48: $C_2\times S_4$

Order 24: $S_4$ x 2, $C_2\times A_4$

Order 16: $C_2\times D_4$

Order 12: $D_6$, $A_4$

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

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

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 48: $C_2\times S_4$

Order 24: $C_2\times A_4$, $S_4$

Order 16: $C_2\times D_4$

Order 12: $D_6$, $A_4$

Order 8: $C_2^3$ x 2, $D_4$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$ x 6, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 48: $C_2\times S_4$ ($C_1$)

Order 24: $S_4$ ($C_2$) x 2, $C_2\times A_4$ ($C_2$)

Order 12: $A_4$ ($C_2^2$)

Order 8: $C_2^3$ ($S_3$)

Order 4: $C_2^2$ ($D_6$)

Order 2: $C_2$ ($S_4$)

Order 1: $C_1$ ($C_2\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 48: $C_2\times S_4$ ($C_1$)

Order 24: $C_2\times A_4$ ($C_2$), $S_4$ ($C_2$)

Order 12: $A_4$ ($C_2^2$)

Order 8: $C_2^3$ ($S_3$)

Order 4: $C_2^2$ ($D_6$)

Order 2: $C_2$ ($S_4$)

Order 1: $C_1$ ($C_2\times S_4$)

Series

Derived series	$C_2\times S_4$	$\rhd$	$A_4$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Chief series	$C_2\times S_4$	$\rhd$	$C_2\times A_4$	$\rhd$	$A_4$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Lower central series	$C_2\times S_4$	$\rhd$	$A_4$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 73 larger groups in the database.

This group is a maximal quotient of 192 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

		1A	2A	2B	2C	2D	2E	3A	4A	4B	6A
	Size	1	1	3	3	6	6	8	6	6	8
	2 P	1A	1A	1A	1A	1A	1A	3A	2C	2C	3A
	3 P	1A	2A	2B	2C	2D	2E	1A	4A	4B	2A
48.48.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.48.1b		$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
48.48.1c		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
48.48.1d		$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$
48.48.2a		$2$	$2$	$2$	$2$	$0$	$0$	$-1$	$0$	$0$	$-1$
48.48.2b		$2$	$-2$	$-2$	$2$	$0$	$0$	$-1$	$0$	$0$	$1$
48.48.3a		$3$	$3$	$-1$	$-1$	$1$	$1$	$0$	$-1$	$-1$	$0$
48.48.3b		$3$	$3$	$-1$	$-1$	$-1$	$-1$	$0$	$1$	$1$	$0$
48.48.3c		$3$	$-3$	$1$	$-1$	$-1$	$1$	$0$	$1$	$-1$	$0$
48.48.3d		$3$	$-3$	$1$	$-1$	$1$	$-1$	$0$	$-1$	$1$	$0$