LMFDB - Non-normal subgroup $D_4:S_3^2 \subseteq D_{20}:S

Subgroup ($H$) information

Description:	$D_4:S_3^2$
Order:	$288$$\medspace = 2^{5} \cdot 3^{2} $
Index:	$5$
Exponent:	$12$$\medspace = 2^{2} \cdot 3 $
Generators:	$a, c^{2}, d^{30}, b, d^{40}, c^{3}, d^{15}$ a, c^{2}, d^{30}, b, d^{40}, c^{3}, d^{15}
Derived length:	$2$

The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), and metabelian.

Ambient group ($G$) information

Description:	$D_{20}:S_3^2$
Order:	$1440$$\medspace = 2^{5} \cdot 3^{2} \cdot 5 $
Exponent:	$60$$\medspace = 2^{2} \cdot 3 \cdot 5 $
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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$(C_6\times S_3\times D_5).C_2^6$
$\operatorname{Aut}(H)$	$C_6^2:D_4^2$, of order $2304$$\medspace = 2^{8} \cdot 3^{2} $
$\operatorname{res}(S)$	$D_4\times D_6^2$, of order $1152$$\medspace = 2^{7} \cdot 3^{2} $
$\card{\operatorname{ker}(\operatorname{res})}$	$4$$\medspace = 2^{2} $
$W$	$D_6^2$, of order $144$$\medspace = 2^{4} \cdot 3^{2} $