Subgroup ($H$) information
| Description: | $C_{55}$ |
| Order: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$c^{44}, c^{20}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{44}.C_{10}^2$ |
| Order: | \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| 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: | $Q_8\times C_{10}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $C_4\times C_2^3:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Outer Automorphisms: | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_{165}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $W$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $Q_8\times C_{55}$ | ||
| Normalizer: | $C_{44}.C_{10}^2$ | ||
| Complements: | $Q_8\times C_{10}$ | ||
| Minimal over-subgroups: | $C_{55}:C_5$ | $C_{110}$ | $C_5\times D_{11}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_5$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $Q_8\times F_{11}$ |