LMFDB - Non-normal subgroup $C_8:C_2^3 \subseteq \GL(2,3):D

Subgroup ($H$) information

Description:	$C_8:C_2^3$
Order:	$64$$\medspace = 2^{6} $
Index:	$15$$\medspace = 3 \cdot 5 $
Exponent:	$8$$\medspace = 2^{3} $
Generators:	$a, b, c^{3}, e^{15}$ a, b, c^{3}, e^{15}
Nilpotency class:	$3$
Derived length:	$2$

The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), metabelian, and rational.

Ambient group ($G$) information

Description:	$\GL(2,3):D_{10}$
Order:	$960$$\medspace = 2^{6} \cdot 3 \cdot 5 $
Exponent:	$120$$\medspace = 2^{3} \cdot 3 \cdot 5 $
Derived length:	$4$

The ambient group is nonabelian and solvable.

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

$\operatorname{Aut}(G)$	$(C_2^2\times D_5\times A_4).C_2^5$
$\operatorname{Aut}(H)$	$C_2^7:C_2^3$, of order $1024$$\medspace = 2^{10} $
$\card{W}$	$16$$\medspace = 2^{4} $