Subgroup ($H$) information
| Description: | $C_4^2:C_{10}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(11\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a^{5}, c^{2}, c, a^{2}, b^{11}, b^{22}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(C_4\times D_{11}):C_{20}$ |
| Order: | \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.(C_{10}\times D_4).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^5.C_2^5$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{res}(S)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $-1$ |
| Projective image | $C_2^2\times F_{11}$ |