Subgroup ($H$) information
| Description: | $C_2\times C_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(7260\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{5}b^{15}c^{3}d^{11}, a^{2}b^{24}, d^{22}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{30}.C_{10}.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $7260$ |
| Number of conjugacy classes in this autjugacy class | $20$ |
| Möbius function | $-2$ |
| Projective image | $C_{11}^2:(S_3\times C_{10}^2)$ |