Abstract group 192.1475: $C_2^2\times \GL(2,3)$

• Label: 192.1475
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $32$
• Rank: $3$

Group information

Description:	$C_2^2\times \GL(2,3)$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3:S_4^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8
Elements	1	55	8	24	56	48	192
Conjugacy classes	1	11	1	4	7	8	32
Divisions	1	11	1	4	7	4	28
Autjugacy classes	1	3	1	2	2	1	10

Dimension	1	2	3	4
Irr. complex chars.	8	12	8	4	32
Irr. rational chars.	8	4	8	8	28

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$32$
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:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=e^{4}=[a,b]=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(3,4)(5,6)(7,8), (1,2)(3,5)(4,6)(7,8)(9,10)(11,12), (9,11)(10,12), (3,6,8)(4,7,5), (1,3,2,5)(4,8,6,7), (1,4,2,6)(3,7,5,8), (1,2)(3,5)(4,6)(7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & -1 & -1 & -1 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 1 & 2 & 1 & 2 \\ 0 & 0 & 1 & 0 \\ 2 & 0 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 2 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 1 & 0 \\ 2 & 0 & 0 & 2 \\ 0 & 0 & 2 & 0 \\ 2 & 2 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 2 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	32T2171				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $\GL(2,3)$
Semidirect product:	$(C_2\times Q_8)$ $\,\rtimes\,$ $D_6$	$Q_8$ $\,\rtimes\,$ $(C_2\times D_6)$	$\SL(2,3)$ $\,\rtimes\,$ $C_2^3$	$(C_2^2\times Q_8)$ $\,\rtimes\,$ $S_3$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $S_4$	$C_2^2$ . $(C_2\times S_4)$	$C_2$ . $(C_2^2\times S_4)$		more information
Aut. group:	$\Aut(C_3\times C_{24})$

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

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

Subgroups

There are 875 subgroups in 213 conjugacy classes, 37 normal (8 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:	$G' \simeq$ $\SL(2,3)$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times Q_8$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^2\times \GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times \SD_{16}$
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: $C_2^2\times \GL(2,3)$

Order 96: $C_2\times \GL(2,3)$ x 6, $C_2^2\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 48: $\GL(2,3)$ x 4, $C_2\times \SL(2,3)$ x 3, $C_2^2\times D_6$

Order 32: $C_2\times \SD_{16}$ x 12, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 24: $C_2\times D_6$ x 14, $\SL(2,3)$, $C_2^2\times C_6$

Order 16: $\SD_{16}$ x 16, $C_2\times D_4$ x 9, $C_2\times C_8$ x 6, $C_2\times Q_8$ x 6, $C_2^4$, $C_2^2\times C_4$

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

Order 8: $C_2^3$ x 11, $D_4$ x 10, $C_2\times C_4$ x 6, $Q_8$ x 5, $C_8$ x 4

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

Order 4: $C_2^2$ x 23, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_2^2\times \GL(2,3)$

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

Order 64: $C_2^2\times \SD_{16}$

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

Order 32: $C_2\times \SD_{16}$ x 2, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 24: $C_2\times D_6$ x 2, $\SL(2,3)$, $C_2^2\times C_6$

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

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

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

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

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_2^2\times \GL(2,3)$ ($C_1$)

Order 96: $C_2\times \GL(2,3)$ ($C_2$) x 6, $C_2^2\times \SL(2,3)$ ($C_2$)

Order 48: $\GL(2,3)$ ($C_2^2$) x 4, $C_2\times \SL(2,3)$ ($C_2^2$) x 3

Order 32: $C_2^2\times Q_8$ ($S_3$)

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

Order 16: $C_2\times Q_8$ ($D_6$) x 3

Order 8: $C_2^3$ ($S_4$), $Q_8$ ($C_2\times D_6$)

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

Order 2: $C_2$ ($C_2\times \GL(2,3)$) x 6, $C_2$ ($C_2^2\times S_4$)

Order 1: $C_1$ ($C_2^2\times \GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_2^2\times \GL(2,3)$ ($C_1$)

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

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

Order 32: $C_2^2\times Q_8$ ($S_3$)

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

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

Order 8: $C_2^3$ ($S_4$), $Q_8$ ($C_2\times D_6$)

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

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

Order 1: $C_1$ ($C_2^2\times \GL(2,3)$)

Series

Derived series

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

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

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

$\rhd$

$\GL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 33 larger groups in the database.

This group is a maximal quotient of 23 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 $28 \times 28$ rational character table.