Subgroup ($H$) information
| Description: | $C_{11}^2:C_5^2$ |
| Order: | \(3025\)\(\medspace = 5^{2} \cdot 11^{2} \) |
| Index: | \(242\)\(\medspace = 2 \cdot 11^{2} \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$a^{22}, bd^{20}, cd^{45}, d^{11}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_5\times C_{11}^3:C_{110}$ |
| Order: | \(732050\)\(\medspace = 2 \cdot 5^{2} \cdot 11^{4} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}.C_{11}^2.C_5^3.C_2^3.C_2$ |
| $\operatorname{Aut}(H)$ | $F_5\times C_{11}^2.C_{10}.\PSL(2,11).C_2$ |
| $W$ | $C_{11}:F_{11}$, of order \(1210\)\(\medspace = 2 \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $1331$ |
| Number of conjugacy classes in this autjugacy class | $11$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^3:C_{110}$ |