Abstract group 192.1537: $C_2^3\times S_4$

• Label: 192.1537
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \cdot 7 \)
• Perm deg.: $10$
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$C_2^3\times S_4$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_4\times C_2^3:\GL(3,2)$, of order \(32256\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	79	8	48	56	192
Conjugacy classes	1	23	1	8	7	40
Divisions	1	23	1	8	7	40
Autjugacy classes	1	4	1	1	1	8

Dimension	1	2	3
Irr. complex chars.	16	8	16	40
Irr. rational chars.	16	8	16	40

Minimal presentations

Permutation degree:	$10$
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	$12285$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{2}=d^{2}=e^{2}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $10$ $\langle(2,3)(5,6)(7,8)(9,10), (9,10), (5,7)(6,8)(9,10), (5,6)(7,8)(9,10), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 1 & -1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 4 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Transitive group:	24T400	32T2168			more information
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $S_4$
Semidirect product:	$C_2^5$ $\,\rtimes\,$ $S_3$	$A_4$ $\,\rtimes\,$ $C_2^4$	$C_2^4$ $\,\rtimes\,$ $D_6$	$(C_2^3\times A_4)$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(\OD_{16}:C_6)$	$\Aut(\SL(2,3):C_8)$	$\Aut(C_8\times \SL(2,3))$	$\Aut(C_8\times S_4)$	all 14

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 2398 subgroups in 701 conjugacy classes, 99 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^3\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
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

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 192: $C_2^3\times S_4$

Order 96: $C_2^2\times S_4$ x 14, $C_2^3\times A_4$

Order 64: $D_4\times C_2^3$

Order 48: $C_2\times S_4$ x 28, $C_2^2\times A_4$ x 7, $C_2^2\times D_6$

Order 32: $C_2^2\times D_4$ x 28, $C_2^5$ x 2, $C_2^3\times C_4$

Order 24: $C_2\times D_6$ x 14, $S_4$ x 8, $C_2\times A_4$ x 7, $C_2^2\times C_6$

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 37, $C_2^2\times C_4$ x 14

Order 12: $D_6$ x 28, $C_2\times C_6$ x 7, $A_4$

Order 8: $C_2^3$ x 134, $D_4$ x 64, $C_2\times C_4$ x 28

Order 6: $S_3$ x 8, $C_6$ x 7

Order 4: $C_2^2$ x 114, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_2^3\times S_4$

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

Order 64: $D_4\times C_2^3$

Order 48: $C_2^2\times D_6$, $C_2^2\times A_4$, $C_2\times S_4$

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

Order 24: $C_2^2\times C_6$, $C_2\times D_6$, $C_2\times A_4$, $S_4$

Order 16: $C_2^4$ x 6, $C_2\times D_4$ 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 9, $D_4$ 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 192: $C_2^3\times S_4$ ($C_1$)

Order 96: $C_2^2\times S_4$ ($C_2$) x 14, $C_2^3\times A_4$ ($C_2$)

Order 48: $C_2\times S_4$ ($C_2^2$) x 28, $C_2^2\times A_4$ ($C_2^2$) x 7

Order 32: $C_2^5$ ($S_3$)

Order 24: $S_4$ ($C_2^3$) x 8, $C_2\times A_4$ ($C_2^3$) x 7

Order 16: $C_2^4$ ($D_6$) x 7

Order 12: $A_4$ ($C_2^4$)

Order 8: $C_2^3$ ($C_2\times D_6$) x 7, $C_2^3$ ($S_4$)

Order 4: $C_2^2$ ($C_2\times S_4$) x 7, $C_2^2$ ($C_2^2\times D_6$)

Order 2: $C_2$ ($C_2^2\times S_4$) x 7

Order 1: $C_1$ ($C_2^3\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_2^3\times S_4$ ($C_1$)

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

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

Order 32: $C_2^5$ ($S_3$)

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

Order 16: $C_2^4$ ($D_6$)

Order 12: $A_4$ ($C_2^4$)

Order 8: $C_2^3$ ($C_2\times D_6$), $C_2^3$ ($S_4$)

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

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

Order 1: $C_1$ ($C_2^3\times S_4$)

Series

Derived series

$C_2^3\times S_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3\times S_4$

$\rhd$

$C_2^3\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^3\times S_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 35 larger groups in the database.

This group is a maximal quotient of 53 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 $40 \times 40$ rational character table. Alternatively, you may search for characters of this group with desired properties.