Normal subgroup $C_3^6.(S_3\times Q_8) \trianglelefteq C_3^6.(S_3\times \GL(2,3))$

• Label: 209952.kc.6.A
• Order: $ 2^{4} \cdot 3^{7} $
• Index: $ 2 \cdot 3 $
• Normal: Yes

Subgroup ($H$) information

Description:	$C_3^6.(S_3\times Q_8)$
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Index:	\(6\)\(\medspace = 2 \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Generators:	$\langle(1,2,3)(7,9,8)(10,11,12)(13,15,14)(19,21,20)(22,23,24)(25,27,26)(31,33,32) \!\cdots\! \rangle$
Derived length:	$3$

The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.

Ambient group ($G$) information

Description:	$C_3^6.(S_3\times \GL(2,3))$
Order:	\(209952\)\(\medspace = 2^{5} \cdot 3^{8} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Derived length:	$5$

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

Quotient group ($Q$) structure

Description:	$S_3$
Order:	\(6\)\(\medspace = 2 \cdot 3 \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism Group:	$S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Outer Automorphisms:	$C_1$, of order $1$
Derived length:	$2$

The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.

Automorphism information

Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3^6.(S_3\times \GL(2,3))$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \)
$\operatorname{Aut}(H)$	$D_6\times \GL(3,2)$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \)
$W$	$C_3^6.(S_3\times \GL(2,3))$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \)

Related subgroups

Centralizer:	$C_1$
Normalizer:	$C_3^6.(S_3\times \GL(2,3))$
Minimal over-subgroups:	$C_3^6:(S_3\times \SL(2,3))$	$C_3^6.C_{12}.C_2^3$
Maximal under-subgroups:	$C_3^6.C_{12}.C_2$	$C_3^5.C_3:S_3.C_2^2$	$C_3^6.C_3.Q_8$	$C_3^6.Q_8.C_2$	$S_3\times C_3^4:Q_8$	$C_3^4.D_6.C_2^2$

Other information

Number of conjugacy classes in this autjugacy class	$1$
Möbius function	not computed
Projective image	$C_3^6.(S_3\times \GL(2,3))$