Subgroup ($H$) information
| Description: | $C_{22}:C_4$ |
| Order: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$ac, c^{4}, d, b^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $(C_2\times C_8).D_{22}$ |
| Order: | \(704\)\(\medspace = 2^{6} \cdot 11 \) |
| Exponent: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^3\times C_{11}:C_5).C_2^6$ |
| $\operatorname{Aut}(H)$ | $D_4\times F_{11}$, of order \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2\times D_{22}$, of order \(88\)\(\medspace = 2^{3} \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_4\times D_{11}$ |