LMFDB - Non-normal subgroup $C_{39}:C_3 \subseteq C_{13}^2:C

Subgroup ($H$) information

Description:	$C_{39}:C_3$
Order:	$117$$\medspace = 3^{2} \cdot 13 $
Index:	$13$
Exponent:	$39$$\medspace = 3 \cdot 13 $
Generators:	$a, bc^{27}, c^{13}$ a, bc^{27}, c^{13}
Derived length:	$2$

The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 3$, and an A-group.

Ambient group ($G$) information

Description:	$C_{13}^2:C_3^2$
Order:	$1521$$\medspace = 3^{2} \cdot 13^{2} $
Exponent:	$39$$\medspace = 3 \cdot 13 $
Derived length:	$2$

The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

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

$\operatorname{Aut}(G)$	$S_3\times C_{13}^2.C_{12}.\PSL(2,13).C_2$
$\operatorname{Aut}(H)$	$S_3\times F_{13}$, of order $936$$\medspace = 2^{3} \cdot 3^{2} \cdot 13 $
$\operatorname{res}(S)$	$S_3\times F_{13}$, of order $936$$\medspace = 2^{3} \cdot 3^{2} \cdot 13 $
$\card{\operatorname{ker}(\operatorname{res})}$	$156$$\medspace = 2^{2} \cdot 3 \cdot 13 $
$W$	$C_{13}:C_3$, of order $39$$\medspace = 3 \cdot 13 $