LMFDB - Non-normal subgroup $C_2^2\times D_4 \subseteq D_{12}:D

Subgroup ($H$) information

Description:	$C_2^2\times D_4$
Order:	$32$$\medspace = 2^{5} $
Index:	$15$$\medspace = 3 \cdot 5 $
Exponent:	$4$$\medspace = 2^{2} $
Generators:	$a, b, c, d^{15}$ a, b, c, d^{15}
Nilpotency class:	$2$
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:	$D_{12}:D_{10}$
Order:	$480$$\medspace = 2^{5} \cdot 3 \cdot 5 $
Exponent:	$60$$\medspace = 2^{2} \cdot 3 \cdot 5 $
Derived length:	$2$

The ambient group 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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{15}:(C_2^2.C_2^6)$
$\operatorname{Aut}(H)$	$C_2^6:(C_2\times S_4)$, of order $3072$$\medspace = 2^{10} \cdot 3 $
$\operatorname{res}(S)$	$C_2^2\times D_4$, of order $32$$\medspace = 2^{5} $
$\card{\operatorname{ker}(\operatorname{res})}$	$8$$\medspace = 2^{3} $
$W$	$C_2^2$, of order $4$$\medspace = 2^{2} $