Subgroup ($H$) information
| Description: | $C_{44}$ |
| Order: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$a, c^{66}, c^{12}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is 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: | $C_{12}.D_{22}$ |
| Order: | \(528\)\(\medspace = 2^{4} \cdot 3 \cdot 11 \) |
| Exponent: | \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{66}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-2$ |
| Projective image | $S_3\times D_{22}$ |