Abstract group 96.226: $C_2^2\times S_4$

• Label: 96.226
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $8$
• Trans deg.: $12$
• Rank: $3$

Group information

Description:	$C_2^2\times S_4$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_4^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Composition factors:	$C_2$ x 5, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	39	8	24	24	96
Conjugacy classes	1	11	1	4	3	20
Divisions	1	11	1	4	3	20
Autjugacy classes	1	4	1	1	1	8

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

Minimal presentations

Permutation degree:	$8$
Transitive degree:	$12$
Rank:	$3$
Inequivalent generating triples:	$420$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=d^{2}=e^{2}=[a,b]=[a,c]=[a,d]=[a,e]=[b,e]=[d,e]=1, c^{b}=c^{5}, d^{b}=de, d^{c}=e, e^{c}=de \rangle$
Permutation group:	$\langle(2,3), (5,6)(7,8), (5,7)(6,8), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrr} 0 & -1 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ -1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 2 & 2 & 1 & 1 \\ 2 & 2 & 1 & 0 \\ 1 & 2 & 2 & 1 \\ 1 & 2 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 2 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 2 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 0 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
	$\left\langle \left(\begin{array}{rr} 5 & 4 \\ 4 & 5 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 3 & 4 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 4 & 3 \\ 1 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/8\Z)$
Transitive group:	12T48	16T182	24T125	24T126	all 8
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $S_4$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $S_3$	$A_4$ $\,\rtimes\,$ $C_2^3$	$C_2^3$ $\,\rtimes\,$ $D_6$	$(C_2^2\times A_4)$ $\,\rtimes\,$ $C_2$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(\OD_{16}:C_2)$	$\Aut(D_4:C_6)$	$\Aut(A_4:C_8)$	$\Aut(\SL(2,3):C_4)$	all 16

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{8}\Z)$.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 420 subgroups in 131 conjugacy classes, 26 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^2\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 96: $C_2^2\times S_4$

Order 48: $C_2\times S_4$ x 6, $C_2^2\times A_4$

Order 32: $C_2^2\times D_4$

Order 24: $S_4$ x 4, $C_2\times A_4$ x 3, $C_2\times D_6$

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 6, $C_2\times C_6$, $A_4$

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $C_2^2\times S_4$

Order 48: $C_2^2\times A_4$, $C_2\times S_4$

Order 32: $C_2^2\times D_4$

Order 24: $C_2\times D_6$, $C_2\times A_4$, $S_4$

Order 16: $C_2^4$ x 2, $C_2\times D_4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$, $D_6$, $A_4$

Order 8: $C_2^3$ x 6, $D_4$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$ x 9, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $C_2^2\times S_4$ ($C_1$)

Order 48: $C_2\times S_4$ ($C_2$) x 6, $C_2^2\times A_4$ ($C_2$)

Order 24: $S_4$ ($C_2^2$) x 4, $C_2\times A_4$ ($C_2^2$) x 3

Order 16: $C_2^4$ ($S_3$)

Order 12: $A_4$ ($C_2^3$)

Order 8: $C_2^3$ ($D_6$) x 3

Order 4: $C_2^2$ ($C_2\times D_6$), $C_2^2$ ($S_4$)

Order 2: $C_2$ ($C_2\times S_4$) x 3

Order 1: $C_1$ ($C_2^2\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $C_2^2\times S_4$ ($C_1$)

Order 48: $C_2^2\times A_4$ ($C_2$), $C_2\times S_4$ ($C_2$)

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

Order 16: $C_2^4$ ($S_3$)

Order 12: $A_4$ ($C_2^3$)

Order 8: $C_2^3$ ($D_6$)

Order 4: $C_2^2$ ($C_2\times D_6$), $C_2^2$ ($S_4$)

Order 2: $C_2$ ($C_2\times S_4$)

Order 1: $C_1$ ($C_2^2\times S_4$)

Series

Derived series

$C_2^2\times S_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^2\times S_4$

$\rhd$

$C_2\times S_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^2\times S_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 52 larger groups in the database.

This group is a maximal quotient of 88 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

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