Subgroup ($H$) information
| Description: | $C_{10}\times \SD_{16}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(121\)\(\medspace = 11^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Generators: |
$a^{5}, d^{22}, d^{11}, a^{2}c^{10}d^{8}, bd^{8}, b^{2}c$
|
| Nilpotency class: | $3$ |
| 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: | $C_{11}^2:(C_{10}\times \SD_{16})$ |
| Order: | \(19360\)\(\medspace = 2^{5} \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_2^3.C_5.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_4^3:C_2^3$, of order \(512\)\(\medspace = 2^{9} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $121$ |
| Möbius function | $-1$ |
| Projective image | $C_2\times D_{11}^2:C_{10}$ |