Subgroup ($H$) information
| Description: | $C_5\times \SD_{64}$ |
| Order: | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| Index: | \(3\) |
| Exponent: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Generators: |
$a, b^{24}, c^{3}, b^{14}, b, b^{16}, b^{4}$
|
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{15}:\SD_{64}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Exponent: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3:((C_2\times C_4\times C_8).C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_4\times D_{16}:C_8$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{res}(S)$ | $C_4\times D_{16}:C_8$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_3:D_{16}$ |