Subgroup ($H$) information
| Description: | $C_2^2\times C_{40}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$a^{5}, c^{20}, c^{5}, a^{2}c^{16}, b^{2}, c^{10}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 2$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_{10}^2:\OD_{16}$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7\times C_4\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_2^4:S_4$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2^5\times C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_{10}\times D_{10}$ |