Subgroup ($H$) information
| Description: | $C_{2490}$ |
| Order: | \(2490\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 83 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(2490\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 83 \) |
| Generators: |
$a^{12450}, a^{16600}, a^{300}, a^{9960}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,83$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{24900}$ |
| Order: | \(24900\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \) |
| Exponent: | \(24900\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 83 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,83$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times C_{820}$, of order \(6560\)\(\medspace = 2^{5} \cdot 5 \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{164}$, of order \(656\)\(\medspace = 2^{4} \cdot 41 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{24900}$ | |||
| Normalizer: | $C_{24900}$ | |||
| Minimal over-subgroups: | $C_{12450}$ | $C_{4980}$ | ||
| Maximal under-subgroups: | $C_{1245}$ | $C_{830}$ | $C_{498}$ | $C_{30}$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{10}$ |