Subgroup ($H$) information
| Description: | $C_5\times D_{22}$ |
| Order: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Index: | \(242\)\(\medspace = 2 \cdot 11^{2} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}d^{11}, a^{2}, b^{2}, b^{11}d^{14}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $D_{11}\times C_{22}:F_{11}$ |
| Order: | \(53240\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{3} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_5.C_{10}^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_{22}:C_{20}$, of order \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| $W$ | $D_{11}$, of order \(22\)\(\medspace = 2 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $484$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $D_{11}\times C_{22}:F_{11}$ |