LMFDB - Normal subgroup $C_3^2 \trianglelefteq C_8:S

Subgroup ($H$) information

Description:	$C_3^2$
Order:	$9$$\medspace = 3^{2} $
Index:	$192$$\medspace = 2^{6} \cdot 3 $
Exponent:	$3$
Generators:	$b^{2}, c^{2}$ b^{2}, c^{2}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.

Ambient group ($G$) information

Description:	$C_8:S_3^3$
Order:	$1728$$\medspace = 2^{6} \cdot 3^{3} $
Exponent:	$24$$\medspace = 2^{3} \cdot 3 $
Derived length:	$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Quotient group ($Q$) structure

Description:	$D_6\times \OD_{16}$
Order:	$192$$\medspace = 2^{6} \cdot 3 $
Exponent:	$24$$\medspace = 2^{3} \cdot 3 $
Automorphism Group:	$S_3\times C_2^5.C_2^4$, of order $3072$$\medspace = 2^{10} \cdot 3 $
Outer Automorphisms:	$C_2\times D_4^2$, of order $128$$\medspace = 2^{7} $
Nilpotency class:	$-1$
Derived length:	$2$

The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Automorphism information

While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3^3.C_2^6.C_2^3$
$\operatorname{Aut}(H)$	$\GL(2,3)$, of order $48$$\medspace = 2^{4} \cdot 3 $
$\operatorname{res}(S)$	$C_2^2$, of order $4$$\medspace = 2^{2} $
$\card{\operatorname{ker}(\operatorname{res})}$	$1728$$\medspace = 2^{6} \cdot 3^{3} $
$W$	$C_2^2$, of order $4$$\medspace = 2^{2} $