LMFDB - Non-normal subgroup $C_{20}.C_2^4 \subseteq C_{60}.C

Subgroup ($H$) information

Description:	$C_{20}.C_2^4$
Order:	$320$$\medspace = 2^{6} \cdot 5 $
Index:	$3$
Exponent:	$40$$\medspace = 2^{3} \cdot 5 $
Generators:	$a, c^{12}, c^{3}, d^{5}, b, d^{2}, c^{6}$ a, c^{12}, c^{3}, d^{5}, b, d^{2}, c^{6}
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:	$C_{60}.C_2^4$
Order:	$960$$\medspace = 2^{6} \cdot 3 \cdot 5 $
Exponent:	$120$$\medspace = 2^{3} \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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{15}:(C_2^3.C_2^6.C_2)$
$\operatorname{Aut}(H)$	$D_{10}.(C_2^4\times S_4)$, of order $7680$$\medspace = 2^{9} \cdot 3 \cdot 5 $
$\operatorname{res}(S)$	$C_2\times F_5\times D_4^2$, of order $2560$$\medspace = 2^{9} \cdot 5 $
$\card{\operatorname{ker}(\operatorname{res})}$	$2$
$W$	$C_2^3:D_{10}$, of order $160$$\medspace = 2^{5} \cdot 5 $