Subgroup ($H$) information
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(10800\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$b^{6}, d^{6}e^{4}$
|
| 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: | $D_5^3.C_3^2:D_6$ |
| Order: | \(108000\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_5^3:\He_3.C_2^3$, of order \(216000\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $150$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $D_5^3.C_3^2:D_6$ |