Properties

Label	1152.137742
Order	\( 2^{7} \cdot 3^{2} \)
Exponent	\( 2^{2} \cdot 3 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{5} \)
$\card{Z(G)}$	\( 2^{2} \)
$\card{\Aut(G)}$	\( 2^{14} \cdot 3^{4} \)
$\card{\mathrm{Out}(G)}$	\( 2^{9} \cdot 3^{2} \)
Trans deg.	not computed
Rank	not computed

Learn more

This group is not stored in the database. However, basic information about the group, computed on the fly, is listed below.

Group information

Description:	$C_3^2 \rtimes (C_2\times Q_8^2)$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order 1327104
Derived length:	$3$

This group is nonabelian and solvable. Whether it is metacyclic, monomial, or rational has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	39	8	984	24	96	1152
Conjugacy classes	1	7	1	42	3	6	60
Divisions	data not computed
Autjugacy classes	data not computed

Dimension	1	2	4	8	16
Irr. complex chars.	32	16	2	8	2	60

Constructions

Presentation:		${\langle a, b, c, d, e, f, g, h, i \mid e^{2}=f^{2}=g^{2}=h^{3}=i^{3}=[a,d]= \!\cdots\! \rangle}$ < a, b, c, d, e, f, g, h, i \| e^2,f^2,g^2,h^3,i^3,[a,d],[a,e],[a,f],[a,g],[b,c],[b,e],[b,f],[b,g],[b,h],[b,i],[c,d],[c,e],[c,f],[c,g],[d,e],[d,f],[d,g],[d,h],[d,i],[e,f],[e,g],[e,h],[e,i],[f,g],[f,h],[f,i],[h,i], fga^-2, fb^-2, gc^-2, fd^-2, bfa^-1b^-1a, cga^-1c^-1a, hia^-1h^-1a, hi^2a^-1i^-1a, dfb^-1d^-1b, i^2c^-1h^-1c, hc^-1i^-1c, h^2g^-1h^-1g, i^2g^-1i^-1*g >

Homology

Abelianization:

Subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_6^2:Q_8$
Commutator:	$G' \simeq$ $C_6:S_3$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3\times \PSU(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_4.C_6^2$	$G/\operatorname{Fit} \simeq$ $Q_8$
Radical:	$R \simeq$ $C_3^2 \rtimes (C_2\times Q_8^2)$	$G/R \simeq$ $C_1$
Socle:	$S \simeq$ $C_6^2$	$G/S \simeq$ $C_2^2\times Q_8$
2-Sylow subgroup:	$P_{2} \simeq$ $C_2\times Q_8^2$
3-Sylow subgroup:	$P_{3} \simeq$ $C_3^2$
Maximal subgroups:	$M_{2,1} \simeq$ $(C_6\times C_{12}):Q_8$	$G/M_{2,1} \simeq$ $C_2$	3 normal subgroups
	$M_{2,2} \simeq$ $(C_6\times C_{12}):Q_8$	$G/M_{2,2} \simeq$ $C_2$	9 normal subgroups
	$M_{2,3} \simeq$ $C_3^2:Q_8^2$	$G/M_{2,3} \simeq$ $C_2$	16 normal subgroups
	$M_{2,4} \simeq$ $C_6^2.(C_2^2\times C_4)$	$G/M_{2,4} \simeq$ $C_2$	3 normal subgroups
	$M_{9} \simeq$ $C_2\times Q_8^2$		9 subgroups in one conjugacy class
Maximal quotients:	$m_{2,1} \simeq$ $C_2$	$G/m_{2,1} \simeq$ $C_3^2:Q_8^2$	2 normal subgroups
	$m_{2,2} \simeq$ $C_2$	$G/m_{2,2} \simeq$ $C_2^3\times \PSU(3,2)$
	$m_{9} \simeq$ $C_3^2$	$G/m_{9} \simeq$ $C_2\times Q_8^2$