Subgroup ($H$) information
| Description: | $C_{11}^3:C_{55}$ |
| Order: | \(73205\)\(\medspace = 5 \cdot 11^{4} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$a^{22}d, b, cd^{45}, a^{10}, d^{5}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}.C_{11}^2.C_5^3.C_2^3.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{11}^3.C_{11}^2.C_{10}^2$ |
| $W$ | $C_{11}^3:C_{110}$, of order \(146410\)\(\medspace = 2 \cdot 5 \cdot 11^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $5$ |
| Number of conjugacy classes in this autjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $C_5\times C_{11}^3:C_{110}$ |