Subgroup ($H$) information
| Description: | $C_{220}:C_5$ |
| Order: | \(1100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Index: | \(121\)\(\medspace = 11^{2} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}, d^{11}, a^{10}, a^{4}, c$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 5$, and an A-group.
Ambient group ($G$) information
| Description: | $C_5\times \He_{11}:C_{20}$ |
| Order: | \(133100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{3} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}.C_{60}.C_{10}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_{110}:C_{20}$, of order \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| $W$ | $C_{11}:C_5$, of order \(55\)\(\medspace = 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $121$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $\He_{11}:C_{20}$ |