Abstract group 192.967: $C_2\times A_4:C_8$

• Label: 192.967
• Order: \( 2^{6} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_2\times A_4:C_8$
Order:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$\GL(2,\mathbb{Z}/4):C_2^2$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 6, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	15	8	16	24	96	32	192
Conjugacy classes	1	7	1	8	3	16	4	40
Divisions	1	7	1	4	3	4	2	22
Autjugacy classes	1	5	1	4	2	2	2	17

Dimension	1	2	3	4	6	12
Irr. complex chars.	16	8	16	0	0	0	40
Irr. rational chars.	4	6	4	4	2	2	22

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$48$
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	4	5	7

Constructions

Presentation:	$\langle a, b, c, d \mid a^{8}=b^{6}=c^{2}=d^{2}=[c,d]=1, b^{a}=b^{5}, c^{a}=d, d^{a}=c, c^{b}=cd, d^{b}=c \rangle$
Permutation group:	Degree $14$ $\langle(2,3)(5,6,7,9,8,10,11,12)(13,14), (13,14), (5,7,8,11)(6,9,10,12), (5,8)(6,10)(7,11)(9,12), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 6 & 15 \\ 1 & 2 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 8 & 15 \end{array}\right), \left(\begin{array}{rr} 3 & 5 \\ 7 & 12 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 4 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Direct product:	$C_2$ $\, \times\, $ $(A_4:C_8)$
Semidirect product:	$(C_2\times A_4)$ $\,\rtimes\,$ $C_8$ (2)	$A_4$ $\,\rtimes\,$ $(C_2\times C_8)$	$C_2^3$ $\,\rtimes\,$ $(C_3:C_8)$ (2)	$C_2^2$ $\,\rtimes\,$ $(C_6:C_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_4)$ . $S_4$	$C_4$ . $(C_2\times S_4)$	$(C_4\times A_4)$ . $C_4$	$C_4$ . $(A_4:C_4)$	all 15

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 254 subgroups in 89 conjugacy classes, 27 normal (19 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:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2\times A_4:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:C_8$
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

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 192: $C_2\times A_4:C_8$

Order 96: $A_4:C_8$ x 2, $C_2^3:C_{12}$

Order 64: $C_2^3:C_8$

Order 48: $C_4\times A_4$ x 2, $C_6:C_8$, $C_2^2\times A_4$

Order 32: $C_2^2:C_8$ x 4, $C_2^2\times C_8$ x 2, $C_2^3\times C_4$

Order 24: $C_2\times A_4$ x 3, $C_3:C_8$ x 2, $C_2\times C_{12}$

Order 16: $C_2\times C_8$ x 8, $C_2^2\times C_4$ x 6, $C_2^4$

Order 12: $C_{12}$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 9, $C_2^3$ x 7, $C_8$ x 4

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 12, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 192: $C_2\times A_4:C_8$

Order 96: $A_4:C_8$, $C_2^3:C_{12}$

Order 64: $C_2^3:C_8$

Order 48: $C_4\times A_4$ x 2, $C_6:C_8$, $C_2^2\times A_4$

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_2^3\times C_4$

Order 24: $C_2\times A_4$ x 2, $C_2\times C_{12}$, $C_3:C_8$

Order 16: $C_2\times C_8$ x 6, $C_2^2\times C_4$ x 6, $C_2^4$

Order 12: $C_{12}$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 9, $C_2^3$ x 5, $C_8$ x 2

Order 6: $C_6$ x 2

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

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 192: $C_2\times A_4:C_8$ ($C_1$)

Order 96: $A_4:C_8$ ($C_2$) x 2, $C_2^3:C_{12}$ ($C_2$)

Order 48: $C_2^2\times A_4$ ($C_4$), $C_4\times A_4$ ($C_2^2$), $C_4\times A_4$ ($C_4$)

Order 32: $C_2^3\times C_4$ ($S_3$)

Order 24: $C_2\times A_4$ ($C_8$) x 2, $C_2\times A_4$ ($C_2\times C_4$)

Order 16: $C_2^4$ ($C_3:C_4$), $C_2^2\times C_4$ ($D_6$), $C_2^2\times C_4$ ($C_3:C_4$)

Order 12: $A_4$ ($C_2\times C_8$)

Order 8: $C_2^3$ ($C_3:C_8$) x 2, $C_2^3$ ($C_6:C_4$), $C_2\times C_4$ ($S_4$)

Order 4: $C_2^2$ ($C_6:C_8$), $C_2^2$ ($A_4:C_4$), $C_4$ ($C_2\times S_4$), $C_4$ ($A_4:C_4$)

Order 2: $C_2$ ($A_4:C_8$) x 2, $C_2$ ($C_2^2.S_4$)

Order 1: $C_1$ ($C_2\times A_4:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 192: $C_2\times A_4:C_8$ ($C_1$)

Order 96: $A_4:C_8$ ($C_2$), $C_2^3:C_{12}$ ($C_2$)

Order 48: $C_2^2\times A_4$ ($C_4$), $C_4\times A_4$ ($C_2^2$), $C_4\times A_4$ ($C_4$)

Order 32: $C_2^3\times C_4$ ($S_3$)

Order 24: $C_2\times A_4$ ($C_2\times C_4$), $C_2\times A_4$ ($C_8$)

Order 16: $C_2^4$ ($C_3:C_4$), $C_2^2\times C_4$ ($D_6$), $C_2^2\times C_4$ ($C_3:C_4$)

Order 12: $A_4$ ($C_2\times C_8$)

Order 8: $C_2^3$ ($C_6:C_4$), $C_2^3$ ($C_3:C_8$), $C_2\times C_4$ ($S_4$)

Order 4: $C_2^2$ ($C_6:C_8$), $C_2^2$ ($A_4:C_4$), $C_4$ ($C_2\times S_4$), $C_4$ ($A_4:C_4$)

Order 2: $C_2$ ($A_4:C_8$), $C_2$ ($C_2^2.S_4$)

Order 1: $C_1$ ($C_2\times A_4:C_8$)

Series

Derived series

$C_2\times A_4:C_8$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2\times A_4:C_8$

$\rhd$

$A_4:C_8$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2\times A_4:C_8$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 44 larger groups in the database.

This group is a maximal quotient of 28 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.