Subgroup ($H$) information
| Description: | $C_{11}^3:C_2$ |
| Order: | \(2662\)\(\medspace = 2 \cdot 11^{3} \) |
| Index: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$a^{55}, cd^{45}, d^{5}, b$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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: | $C_5\times C_{55}$ |
| Order: | \(275\)\(\medspace = 5^{2} \cdot 11 \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Automorphism Group: | $C_{10}\times \GL(2,5)$ |
| Outer Automorphisms: | $C_{10}\times \GL(2,5)$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 5$ (hence hyperelementary), and metacyclic.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\He_{11}.C_{11}^2.C_5^3.C_2^3.C_2$ |
| $\operatorname{Aut}(H)$ | $C_{11}^3.C_{10}.\PSL(3,11)$ |
| $W$ | $C_{11}^3:C_{110}$, of order \(146410\)\(\medspace = 2 \cdot 5 \cdot 11^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-5$ |
| Projective image | $C_5\times C_{11}^3:C_{110}$ |