LMFDB - Normal subgroup $C_{17} \trianglelefteq C_{17}\times C

Subgroup ($H$) information

Description:	$C_{17}$
Order:	$17$
Index:	$238$$\medspace = 2 \cdot 7 \cdot 17 $
Exponent:	$17$
Generators:	$ab^{182}$ ab^{182}
Nilpotency class:	$1$
Derived length:	$1$

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

Ambient group ($G$) information

Description:	$C_{17}\times C_{238}$
Order:	$4046$$\medspace = 2 \cdot 7 \cdot 17^{2} $
Exponent:	$238$$\medspace = 2 \cdot 7 \cdot 17 $
Nilpotency class:	$1$
Derived length:	$1$

The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 17$ (hence hyperelementary), and metacyclic.

Quotient group ($Q$) structure

Description:	$C_{238}$
Order:	$238$$\medspace = 2 \cdot 7 \cdot 17 $
Exponent:	$238$$\medspace = 2 \cdot 7 \cdot 17 $
Automorphism Group:	$C_2\times C_{48}$, of order $96$$\medspace = 2^{5} \cdot 3 $
Outer Automorphisms:	$C_2\times C_{48}$, of order $96$$\medspace = 2^{5} \cdot 3 $
Nilpotency class:	$1$
Derived length:	$1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_6\times C_{16}.\PSL(2,17).C_2$
$\operatorname{Aut}(H)$	$C_{16}$, of order $16$$\medspace = 2^{4} $
$\operatorname{res}(S)$	$C_{16}$, of order $16$$\medspace = 2^{4} $
$\card{\operatorname{ker}(\operatorname{res})}$	$1632$$\medspace = 2^{5} \cdot 3 \cdot 17 $
$W$	$C_1$, of order $1$