LMFDB - Non-normal subgroup $C_8 \subseteq C_8:S

Subgroup ($H$) information

Description:	$C_8$
Order:	$8$$\medspace = 2^{3} $
Index:	$216$$\medspace = 2^{3} \cdot 3^{3} $
Exponent:	$8$$\medspace = 2^{3} $
Generators:	$abcd$ abcd
Nilpotency class:	$1$
Derived length:	$1$

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

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.

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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_3^3.C_2^6.C_2^3$
$\operatorname{Aut}(H)$	$C_2^2$, of order $4$$\medspace = 2^{2} $
$\operatorname{res}(S)$	$C_2^2$, of order $4$$\medspace = 2^{2} $
$\card{\operatorname{ker}(\operatorname{res})}$	$128$$\medspace = 2^{7} $
$W$	$C_2$, of order $2$