Subgroup ($H$) information
| Description: | $C_{11}^2:(Q_8\times C_5^2)$ |
| Order: | \(24200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, cd^{24}, a^{2}d^{22}, b^{6}, b^{15}d^{11}, d^{4}, d^{22}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{110}:F_{11}.D_6$ |
| Order: | \(145200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{30}.C_{10}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2.C_{10}^2.C_{10}.C_2^4$ |
| $W$ | $D_{22}:F_{11}$, of order \(4840\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $1$ |
| Projective image | $C_{11}^2:(C_{10}\times D_6)$ |