LMFDB - Normal subgroup $C_{39} \trianglelefteq C_{13}^2:C

Subgroup ($H$) information

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

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

Quotient group ($Q$) structure

Description:	$C_{13}:C_3$
Order:	$39$$\medspace = 3 \cdot 13 $
Exponent:	$39$$\medspace = 3 \cdot 13 $
Automorphism Group:	$F_{13}$, of order $156$$\medspace = 2^{2} \cdot 3 \cdot 13 $
Outer Automorphisms:	$C_4$, of order $4$$\medspace = 2^{2} $
Nilpotency class:	$-1$
Derived length:	$2$

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

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

$\operatorname{Aut}(G)$	$S_3\times C_{13}^2.C_{12}.\PSL(2,13).C_2$
$\operatorname{Aut}(H)$	$C_2\times C_{12}$, of order $24$$\medspace = 2^{3} \cdot 3 $
$\operatorname{res}(S)$	$C_2\times C_{12}$, of order $24$$\medspace = 2^{3} \cdot 3 $
$\card{\operatorname{ker}(\operatorname{res})}$	$79092$$\medspace = 2^{2} \cdot 3^{2} \cdot 13^{3} $
$W$	$C_3$, of order $3$