Abstract group 384.17948: $C_2^4:S_4$

• Label: 384.17948
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $10$
• Trans deg.: $12$
• Rank: $3$

Group information

Description:	$C_2^4:S_4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4.\GL(2,\mathbb{Z}/4)$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	87	32	168	96	384
Conjugacy classes	1	13	1	10	3	28
Divisions	1	13	1	10	3	28
Autjugacy classes	1	7	1	3	2	14

Dimension	1	2	3	6
Irr. complex chars.	8	4	8	8	28
Irr. rational chars.	8	4	8	8	28

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$12$
Rank:	$3$
Inequivalent generating triples:	$10080$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	6
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{2}=d^{2}=e^{4}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $10$ $\langle(1,2)(3,6)(4,5)(7,8)(9,10), (1,2)(3,7)(4,5)(6,8)(9,10), (9,10), (3,4,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{5}(\F_{2})$
Transitive group:	12T136	12T137	16T724	24T1143	all 22
Direct product:	$C_2$ $\, \times\, $ $(C_2^3:S_4)$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $S_4$	$C_2^5$ $\,\rtimes\,$ $D_6$	$(C_2^3:D_4)$ $\,\rtimes\,$ $S_3$	$(C_2^3:S_4)$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $(C_2\times S_4)$	$C_2^2$ . $(C_2^2\times S_4)$			more information
Aut. group:	$\Aut(C_6\times Q_8)$	$\Aut(C_2\wr C_2^2)$	$\Aut(C_2^3.D_4)$	$\Aut(C_2\wr A_4)$	all 9

Elements of the group are displayed as permutations of degree 10.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 2986 subgroups in 463 conjugacy classes, 28 normal (12 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 384: $C_2^4:S_4$

Order 192: $C_2^3:S_4$ x 4, $C_2^3:S_4$ x 2, $C_2^4:A_4$

Order 128: $C_2^4:D_4$

Order 96: $C_2^2:S_4$ x 4, $C_2^3:A_4$ x 2, $C_2^3:A_4$, $C_2^2\times S_4$

Order 64: $C_2\wr C_2^2$ x 8, $C_2^4:C_4$ x 3, $C_2^3:D_4$ x 3, $D_4:C_2^3$

Order 48: $C_2\times S_4$ x 8, $C_2^2:A_4$, $C_2^2\times A_4$

Order 32: $C_2^2\wr C_2$ x 16, $C_2^3:C_4$ x 12, $D_4:C_2^2$ x 10, $C_2^2\times D_4$ x 8, $C_2^3:C_4$ x 5, $D_4:C_2^2$ x 3, $C_2^5$

Order 24: $S_4$ x 8, $C_2\times A_4$ x 5, $C_2\times D_6$

Order 16: $C_2\times D_4$ x 67, $D_4:C_2$ x 24, $C_2^2:C_4$ x 17, $C_2^4$ x 11, $C_2^2\times C_4$ x 8, $C_2\times Q_8$

Order 12: $D_6$ x 6, $A_4$ x 3, $C_2\times C_6$

Order 8: $D_4$ x 52, $C_2^3$ x 47, $C_2\times C_4$ x 32, $Q_8$ x 4

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

Order 4: $C_2^2$ x 47, $C_4$ x 10

Order 3: $C_3$

Order 2: $C_2$ x 13

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2^4:S_4$

Order 192: $C_2^3:S_4$, $C_2^3:S_4$, $C_2^4:A_4$

Order 128: $C_2^4:D_4$

Order 96: $C_2^3:A_4$, $C_2^3:A_4$, $C_2^2:S_4$, $C_2^2\times S_4$

Order 64: $C_2^4:C_4$ x 2, $C_2^3:D_4$ x 2, $C_2\wr C_2^2$ x 2, $D_4:C_2^3$

Order 48: $C_2\times S_4$ x 3, $C_2^2:A_4$, $C_2^2\times A_4$

Order 32: $C_2^2\times D_4$ x 5, $C_2^2\wr C_2$ x 5, $D_4:C_2^2$ x 3, $C_2^3:C_4$ x 3, $C_2^3:C_4$ x 2, $D_4:C_2^2$ x 2, $C_2^5$

Order 24: $C_2\times A_4$ x 3, $S_4$ x 2, $C_2\times D_6$

Order 16: $C_2\times D_4$ x 22, $C_2^4$ x 8, $C_2^2:C_4$ x 6, $C_2^2\times C_4$ x 5, $D_4:C_2$ x 4, $C_2\times Q_8$

Order 12: $D_6$ x 2, $A_4$ x 2, $C_2\times C_6$

Order 8: $C_2^3$ x 26, $D_4$ x 12, $C_2\times C_4$ x 11, $Q_8$

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

Order 4: $C_2^2$ x 24, $C_4$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2^4:S_4$ ($C_1$)

Order 192: $C_2^3:S_4$ ($C_2$) x 4, $C_2^3:S_4$ ($C_2$) x 2, $C_2^4:A_4$ ($C_2$)

Order 96: $C_2^2:S_4$ ($C_2^2$) x 4, $C_2^3:A_4$ ($C_2^2$) x 2, $C_2^3:A_4$ ($C_2^2$)

Order 64: $C_2^3:D_4$ ($S_3$)

Order 48: $C_2^2:A_4$ ($C_2^3$)

Order 32: $C_2^2\wr C_2$ ($D_6$) x 2, $C_2^5$ ($D_6$)

Order 16: $C_2^4$ ($C_2\times D_6$), $C_2^4$ ($S_4$)

Order 8: $C_2^3$ ($C_2\times S_4$) x 3

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

Order 2: $C_2$ ($C_2^3:S_4$)

Order 1: $C_1$ ($C_2^4:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2^4:S_4$ ($C_1$)

Order 192: $C_2^3:S_4$ ($C_2$), $C_2^3:S_4$ ($C_2$), $C_2^4:A_4$ ($C_2$)

Order 96: $C_2^3:A_4$ ($C_2^2$), $C_2^3:A_4$ ($C_2^2$), $C_2^2:S_4$ ($C_2^2$)

Order 64: $C_2^3:D_4$ ($S_3$)

Order 48: $C_2^2:A_4$ ($C_2^3$)

Order 32: $C_2^5$ ($D_6$), $C_2^2\wr C_2$ ($D_6$)

Order 16: $C_2^4$ ($C_2\times D_6$), $C_2^4$ ($S_4$)

Order 8: $C_2^3$ ($C_2\times S_4$) x 2

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

Order 2: $C_2$ ($C_2^3:S_4$)

Order 1: $C_1$ ($C_2^4:S_4$)

Series

Derived series

$C_2^4:S_4$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^4:S_4$

$\rhd$

$C_2^3:S_4$

$\rhd$

$C_2^2:S_4$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^4:S_4$

$\rhd$

$C_2^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 41 larger groups in the database.

This group is a maximal quotient of 65 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 $28 \times 28$ rational character table. Alternatively, you may search for characters of this group with desired properties.