LMFDB - Normal subgroup $C_{39} \trianglelefteq C_{13}\times C

Subgroup ($H$) information

Description:	$C_{39}$
Order:	$39$$\medspace = 3 \cdot 13 $
Index:	$13$
Exponent:	$39$$\medspace = 3 \cdot 13 $
Generators:	$b^{13}, ab^{6}$ b^{13}, ab^{6}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is normal, maximal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,13$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.

Ambient group ($G$) information

Description:	$C_{13}\times C_{39}$
Order:	$507$$\medspace = 3 \cdot 13^{2} $
Exponent:	$39$$\medspace = 3 \cdot 13 $
Nilpotency class:	$1$
Derived length:	$1$

The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 13$ (hence hyperelementary), and metacyclic.

Quotient group ($Q$) structure

Description:	$C_{13}$
Order:	$13$
Exponent:	$13$
Automorphism Group:	$C_{12}$, of order $12$$\medspace = 2^{2} \cdot 3 $
Outer Automorphisms:	$C_{12}$, of order $12$$\medspace = 2^{2} \cdot 3 $
Nilpotency class:	$1$
Derived length:	$1$

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

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

$\operatorname{Aut}(G)$	$C_2\times C_{12}.\PSL(2,13).C_2$, of order $52416$$\medspace = 2^{6} \cdot 3^{2} \cdot 7 \cdot 13 $
$\operatorname{Aut}(H)$	$C_2\times C_{12}$, of order $24$$\medspace = 2^{3} \cdot 3 $
$\operatorname{res}(S)$	$C_2\times C_{12}$, of order $24$$\medspace = 2^{3} \cdot 3 $
$\card{\operatorname{ker}(\operatorname{res})}$	$156$$\medspace = 2^{2} \cdot 3 \cdot 13 $
$W$	$C_1$, of order $1$