LMFDB - Normal subgroup $C_{239}:C_{17} \trianglelefteq C_{4063}:C

Subgroup ($H$) information

Description:	$C_{239}:C_{17}$
Order:	$4063$$\medspace = 17 \cdot 239 $
Index:	$34$$\medspace = 2 \cdot 17 $
Exponent:	$4063$$\medspace = 17 \cdot 239 $
Generators:	$ab^{8}, b^{34}$ ab^{8}, b^{34}
Derived length:	$2$

The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 17$.

Ambient group ($G$) information

Description:	$C_{4063}:C_{34}$
Order:	$138142$$\medspace = 2 \cdot 17^{2} \cdot 239 $
Exponent:	$8126$$\medspace = 2 \cdot 17 \cdot 239 $
Derived length:	$2$

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

Quotient group ($Q$) structure

Description:	$C_{34}$
Order:	$34$$\medspace = 2 \cdot 17 $
Exponent:	$34$$\medspace = 2 \cdot 17 $
Automorphism Group:	$C_{16}$, of order $16$$\medspace = 2^{4} $
Outer Automorphisms:	$C_{16}$, of order $16$$\medspace = 2^{4} $
Derived length:	$1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,17$), 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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{4063}.C_{476}.C_2^2.C_2$
$\operatorname{Aut}(H)$	$F_{239}$, of order $56882$$\medspace = 2 \cdot 7 \cdot 17 \cdot 239 $
$W$	$C_{239}:C_{17}$, of order $4063$$\medspace = 17 \cdot 239 $