LMFDB - Normal subgroup $C_{155} \trianglelefteq C_{31}\times C

Subgroup ($H$) information

Description:	$C_{155}$
Order:	$155$$\medspace = 5 \cdot 31 $
Index:	$31$
Exponent:	$155$$\medspace = 5 \cdot 31 $
Generators:	$b^{31}, ab^{60}$ b^{31}, ab^{60}
Nilpotency class:	$1$
Derived length:	$1$

The subgroup is normal, maximal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,31$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.

Ambient group ($G$) information

Description:	$C_{31}\times C_{155}$
Order:	$4805$$\medspace = 5 \cdot 31^{2} $
Exponent:	$155$$\medspace = 5 \cdot 31 $
Nilpotency class:	$1$
Derived length:	$1$

The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 31$ (hence hyperelementary), and metacyclic.

Quotient group ($Q$) structure

Description:	$C_{31}$
Order:	$31$
Exponent:	$31$
Automorphism Group:	$C_{30}$, of order $30$$\medspace = 2 \cdot 3 \cdot 5 $
Outer Automorphisms:	$C_{30}$, of order $30$$\medspace = 2 \cdot 3 \cdot 5 $
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 $W = N_G(H) / Z_G(H)$.

$\operatorname{Aut}(G)$	$C_4\times C_{30}.\PSL(2,31).C_2$
$\operatorname{Aut}(H)$	$C_2\times C_{60}$, of order $120$$\medspace = 2^{3} \cdot 3 \cdot 5 $
$\operatorname{res}(S)$	$C_2\times C_{60}$, of order $120$$\medspace = 2^{3} \cdot 3 \cdot 5 $
$\card{\operatorname{ker}(\operatorname{res})}$	$930$$\medspace = 2 \cdot 3 \cdot 5 \cdot 31 $
$W$	$C_1$, of order $1$