LMFDB - Non-normal subgroup $D_4\times D_{10} \subseteq D_{12}:D

Subgroup ($H$) information

Description:	$D_4\times D_{10}$
Order:	$160$$\medspace = 2^{5} \cdot 5 $
Index:	$3$
Exponent:	$20$$\medspace = 2^{2} \cdot 5 $
Generators:	$a, d^{12}, c, d^{15}, b, d^{30}$ a, d^{12}, c, d^{15}, b, d^{30}
Derived length:	$2$

The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

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

$\operatorname{Aut}(G)$	$C_{15}:(C_2^2.C_2^6)$
$\operatorname{Aut}(H)$	$F_5\times C_2^5:D_4$, of order $5120$$\medspace = 2^{10} \cdot 5 $
$\operatorname{res}(S)$	$D_{10}.C_2^5$, of order $640$$\medspace = 2^{7} \cdot 5 $
$\card{\operatorname{ker}(\operatorname{res})}$	$2$
$W$	$C_2\times D_{10}$, of order $40$$\medspace = 2^{3} \cdot 5 $