Abstract group 192.951: $C_4\times \GL(2,3)$

• Label: 192.951
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $12$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$C_4\times \GL(2,3)$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^4\times S_4$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	27	8	52	24	48	32	192
Conjugacy classes	1	5	1	10	3	8	4	32
Divisions	1	5	1	6	3	3	2	21
Autjugacy classes	1	4	1	5	3	2	1	17

Dimension	1	2	3	4	6	8
Irr. complex chars.	8	12	8	4	0	0	32
Irr. rational chars.	4	4	4	5	2	2	21

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	6

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{12}=d^{4}=1, c^{2}=d^{2}, b^{a}=b^{5}, c^{a}=cd, d^{a}=d^{3}, c^{b}=cd^{3}, d^{b}=cd^{2}, d^{c}=d^{3} \rangle$
Permutation group:	Degree $12$ $\langle(1,2)(3,6)(4,5), (1,2)(3,5)(4,6)(7,8)(9,10,11,12), (9,11)(10,12), (3,6,8)(4,7,5), (1,3,2,5)(4,8,6,7), (1,4,2,6)(3,7,5,8), (1,2)(3,5)(4,6)(7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 2 & 0 & 0 & 2 \\ 1 & 2 & 0 & 1 \\ 1 & 0 & 2 & 0 \\ 2 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 2 \\ 1 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 0 & 0 & 2 \\ 0 & 2 & 2 & 1 \\ 1 & 2 & 1 & 2 \\ 1 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 2 & 2 & 0 \\ 0 & 2 & 1 & 2 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 1 \\ 2 & 0 & 2 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	32T2167				more information
Direct product:	$C_4$ $\, \times\, $ $\GL(2,3)$
Semidirect product:	$(C_4\times Q_8)$ $\,\rtimes\,$ $S_3$	$Q_8$ $\,\rtimes\,$ $(C_4\times S_3)$	$(C_4\times \SL(2,3))$ $\,\rtimes\,$ $C_2$	$(\SL(2,3):C_4)$ $\,\rtimes\,$ $C_2$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_4)$ . $S_4$	$C_2$ . $(C_4\times S_4)$	$(C_2\times Q_8)$ . $D_6$	$C_2^2$ . $(C_2\times S_4)$	all 8
Aut. group:	$\Aut(C_3\times C_{15})$	$\Aut(C_3\times C_{30})$

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 343 subgroups in 88 conjugacy classes, 19 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $\SL(2,3)$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_4\times Q_8$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4\times \GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times \SD_{16}$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 192: $C_4\times \GL(2,3)$

Order 96: $C_4\times \SL(2,3)$, $\SL(2,3):C_4$, $C_2\times \GL(2,3)$

Order 64: $C_4\times \SD_{16}$

Order 48: $\GL(2,3)$ x 2, $C_4\times D_6$, $C_2\times \SL(2,3)$

Order 32: $D_4:C_4$, $C_2\times \SD_{16}$, $C_4\times C_8$, $C_4\times Q_8$, $C_4\times D_4$, $C_8:C_4$, $Q_8:C_4$

Order 24: $C_4\times S_3$ x 4, $C_2\times C_{12}$, $C_6:C_4$, $\SL(2,3)$, $C_2\times D_6$

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$, $C_4^2$, $C_2\times Q_8$, $C_2\times D_4$, $C_2^2\times C_4$

Order 12: $D_6$ x 6, $C_{12}$ x 2, $C_3:C_4$ x 2, $C_2\times C_6$

Order 8: $C_2\times C_4$ x 6, $D_4$ x 3, $C_8$ x 3, $Q_8$ x 2, $C_2^3$

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

Order 4: $C_4$ x 6, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_4\times \GL(2,3)$

Order 96: $C_4\times \SL(2,3)$, $\SL(2,3):C_4$, $C_2\times \GL(2,3)$

Order 64: $C_4\times \SD_{16}$

Order 48: $C_4\times D_6$, $C_2\times \SL(2,3)$, $\GL(2,3)$

Order 32: $D_4:C_4$, $C_2\times \SD_{16}$, $C_4\times C_8$, $C_4\times Q_8$, $C_4\times D_4$, $C_8:C_4$, $Q_8:C_4$

Order 24: $C_2\times C_{12}$, $C_6:C_4$, $C_4\times S_3$, $\SL(2,3)$, $C_2\times D_6$

Order 16: $\SD_{16}$ x 2, $C_2\times C_8$ x 2, $C_4:C_4$ x 2, $C_2^2:C_4$, $C_4^2$, $C_2\times Q_8$, $C_2\times D_4$, $C_2^2\times C_4$

Order 12: $D_6$ x 3, $C_2\times C_6$, $C_{12}$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 5, $Q_8$ x 2, $D_4$ x 2, $C_8$ x 2, $C_2^3$

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

Order 4: $C_4$ x 5, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_4\times \GL(2,3)$ ($C_1$)

Order 96: $C_4\times \SL(2,3)$ ($C_2$), $\SL(2,3):C_4$ ($C_2$), $C_2\times \GL(2,3)$ ($C_2$)

Order 48: $\GL(2,3)$ ($C_4$) x 2, $C_2\times \SL(2,3)$ ($C_2^2$)

Order 32: $C_4\times Q_8$ ($S_3$)

Order 24: $\SL(2,3)$ ($C_2\times C_4$)

Order 16: $C_2\times Q_8$ ($D_6$)

Order 8: $Q_8$ ($C_4\times S_3$), $C_2\times C_4$ ($S_4$)

Order 4: $C_4$ ($\GL(2,3)$) x 2, $C_2^2$ ($C_2\times S_4$)

Order 2: $C_2$ ($\GL(2,3):C_2$), $C_2$ ($C_2\times \GL(2,3)$), $C_2$ ($C_4\times S_4$)

Order 1: $C_1$ ($C_4\times \GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_4\times \GL(2,3)$ ($C_1$)

Order 96: $C_4\times \SL(2,3)$ ($C_2$), $\SL(2,3):C_4$ ($C_2$), $C_2\times \GL(2,3)$ ($C_2$)

Order 48: $C_2\times \SL(2,3)$ ($C_2^2$), $\GL(2,3)$ ($C_4$)

Order 32: $C_4\times Q_8$ ($S_3$)

Order 24: $\SL(2,3)$ ($C_2\times C_4$)

Order 16: $C_2\times Q_8$ ($D_6$)

Order 8: $Q_8$ ($C_4\times S_3$), $C_2\times C_4$ ($S_4$)

Order 4: $C_2^2$ ($C_2\times S_4$), $C_4$ ($\GL(2,3)$)

Order 2: $C_2$ ($\GL(2,3):C_2$), $C_2$ ($C_2\times \GL(2,3)$), $C_2$ ($C_4\times S_4$)

Order 1: $C_1$ ($C_4\times \GL(2,3)$)

Series

Derived series

$C_4\times \GL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_4\times \GL(2,3)$

$\rhd$

$C_2\times \GL(2,3)$

$\rhd$

$C_2\times \SL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_4\times \GL(2,3)$

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 64 larger groups in the database.

This group is a maximal quotient of 25 larger groups in the database.

Character theory

Complex character table

See the $32 \times 32$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.