Subgroup ($H$) information
| Description: | $C_{30}$ |
| Order: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Index: | \(4965\)\(\medspace = 3 \cdot 5 \cdot 331 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a^{15}, b^{993}, b^{3310}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, and cyclic (hence elementary ($p = 2,3,5$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_{4965}:C_{30}$ |
| Order: | \(148950\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \cdot 331 \) |
| Exponent: | \(9930\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 331 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{331}:C_{15}$ |
| Order: | \(4965\)\(\medspace = 3 \cdot 5 \cdot 331 \) |
| Exponent: | \(4965\)\(\medspace = 3 \cdot 5 \cdot 331 \) |
| Automorphism Group: | $F_{331}$, of order \(109230\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 331 \) |
| Outer Automorphisms: | $C_{22}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4965}.C_{330}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-331$ |
| Projective image | $C_{331}:C_{15}$ |