Subgroup ($H$) information
| Description: | $C_{22}$ |
| Order: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Index: | \(23276\)\(\medspace = 2^{2} \cdot 11 \cdot 23^{2} \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$b^{11}c^{198}d^{9}, c^{46}$
|
| 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: | $F_{23}\wr C_2$ |
| Order: | \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| Exponent: | \(1012\)\(\medspace = 2^{2} \cdot 11 \cdot 23 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{23}\wr C_2$, of order \(512072\)\(\medspace = 2^{3} \cdot 11^{2} \cdot 23^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $1058$ |
| Möbius function | $0$ |
| Projective image | $F_{23}\wr C_2$ |