Subgroup ($H$) information
| Description: | $C_{20}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(2420\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$c^{55}, c^{110}, c^{44}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{55}:(Q_8\times F_{11})$ |
| Order: | \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{11}:F_{11}$ |
| Order: | \(2420\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $F_{11}\wr C_2$, of order \(24200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{2} \) |
| Outer Automorphisms: | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{10}^2.C_{10}.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{220}:F_{11}$ | ||||
| Normalizer: | $C_{55}:(Q_8\times F_{11})$ | ||||
| Minimal over-subgroups: | $C_{220}$ | $C_{220}$ | $C_5\times C_{20}$ | $C_2\times C_{20}$ | $C_5\times Q_8$ |
| Maximal under-subgroups: | $C_{10}$ | $C_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-242$ |
| Projective image | $D_{22}:F_{11}$ |