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Properties

Label	1536.408556859
Order	\( 2^{9} \cdot 3 \)
Exponent	\( 2^{4} \cdot 3 \)

Nilpotent	no
Solvable	yes
$\card{G^{\mathrm{ab}}}$	\( 2^{6} \)
$\card{Z(G)}$	\( 2^{6} \)
$\card{\Aut(G)}$	\( 2^{13} \cdot 3 \)
$\card{\mathrm{Out}(G)}$	\( 2^{10} \)
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:	$\SL(2,3) \rtimes (C_2^2\times C_{16})$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	Group of order 24576
Derived length:	$4$

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	8	12	16	24	48
Elements	1	55	8	104	56	352	64	512	128	256	1536
Conjugacy classes	1	11	1	20	7	64	8	96	16	32	256
Divisions	data not computed
Autjugacy classes	data not computed

Dimension	1	2	3	4
Irr. complex chars.	64	96	64	32	256

Constructions

Presentation:		${\langle a, b, c, d, e, f, g, h, i, j \mid a^{2}=b^{3}=c^{2}=g^{2}=j^{2}= \!\cdots\! \rangle}$ < a, b, c, d, e, f, g, h, i, j \| a^2,b^3,c^2,g^2,j^2,[a,c],[a,d],[a,g],[a,h],[a,i],[a,j],[b,c],[b,d],[b,g],[b,h],[b,i],[b,j],[c,d],[c,e],[c,f],[c,g],[c,h],[c,i],[c,j],[d,e],[d,f],[d,g],[d,h],[d,i],[d,j],[e,g],[e,h],[e,i],[e,j],[f,g],[f,h],[f,i],[f,j],[g,h],[g,i],[g,j],[h,i],[h,j],[i,j], hd^-2, ge^-2, gf^-2, ih^-2, ji^-2, b^2a^-1b^-1a, efa^-1e^-1a, fga^-1f^-1a, fb^-1e^-1b, efgb^-1f^-1b, fge^-1f^-1e >
Aut. group:	$\Aut(C_3\times C_{192})$	$\Aut(C_3\times C_{204})$

Homology

Abelianization:

$C_{2}^{2} \times C_{16} $

Subgroups

Center:	$Z \simeq$ $C_2^2\times C_{16}$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $\SL(2,3)$	$G/G' \simeq$ $C_2^2\times C_{16}$
Frattini:	$\Phi \simeq$ $C_2\times C_8$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times Q_8\times C_{16}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $\SL(2,3) \rtimes (C_2^2\times C_{16})$	$G/R \simeq$ $C_1$
Socle:	$S \simeq$ $C_2^3$	$G/S \simeq$ $C_8\times S_4$
2-Sylow subgroup:	$P_{2} \simeq$ $(C_4\times C_8) . C_2^4$
3-Sylow subgroup:	$P_{3} \simeq$ $C_3$
Maximal subgroups:	$M_{2,1} \simeq$ $(C_2\times Q_8\times C_{16}) \rtimes C_3$	$G/M_{2,1} \simeq$ $C_2$
	$M_{2,2} \simeq$ $\SL(2,3) \rtimes (C_2\times C_{16})$	$G/M_{2,2} \simeq$ $C_2$
	$M_{2,3} \simeq$ $C_{16}\times \GL(2,3)$	$G/M_{2,3} \simeq$ $C_2$	4 normal subgroups
	$M_{2,4} \simeq$ $\SL(2,3) \rtimes (C_2^2\times C_8)$	$G/M_{2,4} \simeq$ $C_2$
	$M_{3} \simeq$ $(C_4\times C_8) . C_2^4$		3 subgroups in one conjugacy class
	$M_{4} \simeq$ $C_{48}:C_2^3$		4 subgroups in one conjugacy class
Maximal quotients:	$m_{2,1} \simeq$ $C_2$	$G/m_{2,1} \simeq$ $C_{16}\times \GL(2,3)$	4 normal subgroups
	$m_{2,2} \simeq$ $C_2$	$G/m_{2,2} \simeq$ $\SL(2,3) \rtimes (C_2^2\times C_8)$
	$m_{2,3} \simeq$ $C_2$	$G/m_{2,3} \simeq$ $(C_2\times C_{16}) \times S_4$
	$m_{2,4} \simeq$ $C_2$	$G/m_{2,4} \simeq$ $\SL(2,3) . (C_2^2\times C_8)$