Subgroup ($H$) information
| Description: | $C_{120}$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$a^{7140}, a^{45696}, a^{14280}, a^{19040}, a^{28560}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{57120}$ |
| Order: | \(57120\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
| Exponent: | \(57120\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_{476}$ |
| Order: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Exponent: | \(476\)\(\medspace = 2^{2} \cdot 7 \cdot 17 \) |
| Automorphism Group: | $C_2^2\times C_{48}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2\times C_{48}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^3\times C_4\times C_8\times C_{48}$ |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{W}$ | $1$ |
Related subgroups
| Centralizer: | $C_{57120}$ | ||
| Normalizer: | $C_{57120}$ | ||
| Minimal over-subgroups: | $C_{2040}$ | $C_{840}$ | $C_{240}$ |
| Maximal under-subgroups: | $C_{60}$ | $C_{40}$ | $C_{24}$ |
Other information
| Möbius function | not computed |
| Projective image | not computed |