Subgroup ($H$) information
| Description: | $C_{80}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Generators: |
$b, b^{2}, b^{4}, c^{3}, b^{8}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{60}.D_8$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:(C_4.C_2^6.C_2^2)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4^2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $C_2\times C_4^2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_{80}$ | |
| Normalizer: | $C_{16}.C_{20}$ | |
| Normal closure: | $C_3:C_{80}$ | |
| Core: | $C_{40}$ | |
| Minimal over-subgroups: | $C_3:C_{80}$ | $C_2\times C_{80}$ |
| Maximal under-subgroups: | $C_{40}$ | $C_{16}$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $C_6.D_8$ |