LMFDB - Normal subgroup $C_{10} \trianglelefteq C_{10}\times C

Subgroup ($H$) information

Description:	$C_{10}$
Order:	$10$$\medspace = 2 \cdot 5 $
Index:	$430$$\medspace = 2 \cdot 5 \cdot 43 $
Exponent:	$10$$\medspace = 2 \cdot 5 $
Generators:	$a^{5}b^{215}, a^{2}$ a^{5}b^{215}, a^{2}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.

Ambient group ($G$) information

Description:	$C_{10}\times C_{430}$
Order:	$4300$$\medspace = 2^{2} \cdot 5^{2} \cdot 43 $
Exponent:	$430$$\medspace = 2 \cdot 5 \cdot 43 $
Nilpotency class:	$1$
Derived length:	$1$

The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.

Quotient group ($Q$) structure

Description:	$C_{430}$
Order:	$430$$\medspace = 2 \cdot 5 \cdot 43 $
Exponent:	$430$$\medspace = 2 \cdot 5 \cdot 43 $
Automorphism Group:	$C_2\times C_{84}$, of order $168$$\medspace = 2^{3} \cdot 3 \cdot 7 $
Outer Automorphisms:	$C_2\times C_{84}$, of order $168$$\medspace = 2^{3} \cdot 3 \cdot 7 $
Nilpotency class:	$1$
Derived length:	$1$

The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,43$), 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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_{42}\times S_3\times \GL(2,5)$
$\operatorname{Aut}(H)$	$C_4$, of order $4$$\medspace = 2^{2} $
$\operatorname{res}(S)$	$C_4$, of order $4$$\medspace = 2^{2} $
$\card{\operatorname{ker}(\operatorname{res})}$	$1680$$\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
$W$	$C_1$, of order $1$