Subgroup ($H$) information
| Description: | $C_{60}$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(305\)\(\medspace = 5 \cdot 61 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$a^{15}, a^{12}b^{61}, a^{30}, a^{40}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_5\times F_{61}$ |
| Order: | \(18300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 61 \) |
| Exponent: | \(3660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 61 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{305}.C_{15}.C_4^2$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $61$ |
| Möbius function | $1$ |
| Projective image | $C_5\times F_{61}$ |