Subgroup ($H$) information
| Description: | $C_{11}^2:D_6$ |
| Order: | \(1452\)\(\medspace = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Index: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$a^{5}, b^{20}c^{4}d^{36}, cd^{40}, d^{22}, d^{4}$
|
| Derived length: | $3$ |
The subgroup is normal, a semidirect factor, nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_{220}:F_{11}:S_3$ |
| 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.
Quotient group ($Q$) structure
| Description: | $C_{10}^2$ |
| Order: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $S_3\times \GL(2,5)$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Outer Automorphisms: | $S_3\times \GL(2,5)$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{30}.C_{10}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{11}^2:(C_{10}\times D_6)$, of order \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| $W$ | $C_{11}^2:(C_{10}\times D_6)$, of order \(14520\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $10$ |
| Projective image | $C_{11}^2:(S_3\times C_{10}^2)$ |