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

Subgroup ($H$) information

Description:	$C_3$
Order:	$3$
Index:	$507$$\medspace = 3 \cdot 13^{2} $
Exponent:	$3$
Generators:	$ac^{2}$ ac^{2}
Nilpotency class:	$1$
Derived length:	$1$

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

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

$\operatorname{Aut}(G)$	$S_3\times C_{13}^2.C_{12}.\PSL(2,13).C_2$
$\operatorname{Aut}(H)$	$C_2$, of order $2$
$\operatorname{res}(S)$	$C_1$, of order $1$
$\card{\operatorname{ker}(\operatorname{res})}$	$52416$$\medspace = 2^{6} \cdot 3^{2} \cdot 7 \cdot 13 $
$W$	$C_1$, of order $1$