Subgroup ($H$) information
| Description: | $C_{31}$ |
| Order: | \(31\) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(31\) |
| Generators: |
$a^{10}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $31$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{310}$ |
| Order: | \(310\)\(\medspace = 2 \cdot 5 \cdot 31 \) |
| Exponent: | \(310\)\(\medspace = 2 \cdot 5 \cdot 31 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,31$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{60}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{30}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{310}$ | |
| Normalizer: | $C_{310}$ | |
| Complements: | $C_{10}$ | |
| Minimal over-subgroups: | $C_{155}$ | $C_{62}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{10}$ |