Subgroup ($H$) information
| Description: | $C_9\times D_5$ |
| Order: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Index: | \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(1,9,2,3,7,4,6,8,5)(10,12,11,13,14), (10,14,13,11,12), (1,3,6)(2,4,5)(7,8,9)(10,11)(13,14), (11,13)(12,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $A_5\times {}^2G(2,3)$ |
| Order: | \(90720\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(630\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times {}^2G(2,3)$, of order \(181440\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6\times F_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $W$ | $C_3\times D_{10}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $168$ |
| Möbius function | $0$ |
| Projective image | $A_5\times {}^2G(2,3)$ |