Subgroup ($H$) information
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$b^{10}c^{22}, c^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $(C_2\times C_{88}):C_{10}$ |
| Order: | \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_5\times \SD_{16}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Automorphism Group: | $C_4^2:C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The quotient is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.(C_2^5\times C_{10})$ |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(704\)\(\medspace = 2^{6} \cdot 11 \) |
| $W$ | $C_5$, of order \(5\) |
Related subgroups
| Centralizer: | $D_4:C_{44}$ | ||
| Normalizer: | $(C_2\times C_{88}):C_{10}$ | ||
| Minimal over-subgroups: | $C_{11}:C_{10}$ | $C_2\times C_{22}$ | $C_2\times C_{22}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{88}:C_{10}$ |