Subgroup ($H$) information
| Description: | $C_{11}:D_4$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$a^{5}, c^{22}, c^{4}, b^{5}c^{33}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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: | $C_5\times C_{10}$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{110}.C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-5$ |
| Projective image | $C_{22}:C_{10}^2$ |