LMFDB - Non-normal subgroup $C_7^3 \subseteq D_7\times C_7^3:S

Subgroup ($H$) information

Description:	$C_7^3$
Order:	$343$$\medspace = 7^{3} $
Index:	$336$$\medspace = 2^{4} \cdot 3 \cdot 7 $
Exponent:	$7$
Generators:	$d^{2}, ef^{2}, c^{2}f$ d^{2}, ef^{2}, c^{2}f
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).

Ambient group ($G$) information

Description:	$D_7\times C_7^3:S_4$
Order:	$115248$$\medspace = 2^{4} \cdot 3 \cdot 7^{4} $
Exponent:	$84$$\medspace = 2^{2} \cdot 3 \cdot 7 $
Derived length:	$4$

The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.

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

$\operatorname{Aut}(G)$	$C_7^3.(C_7\times A_4).C_6^2.C_2$
$\operatorname{Aut}(H)$	$\GL(3,7)$, of order $33784128$$\medspace = 2^{6} \cdot 3^{4} \cdot 7^{3} \cdot 19 $
$W$	$C_2^2$, of order $4$$\medspace = 2^{2} $