Subgroup ($H$) information
| Description: | $D_{15}$ |
| Order: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$a, d^{2}, b$
|
| 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: | $C_5^2:S_4$ |
| Order: | \(600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(S_4\times C_5^2):\GL(2,5)$, of order \(288000\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_3\times F_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $S_3\times F_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $D_{15}$, of order \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $20$ |
| Möbius function | $1$ |
| Projective image | $C_5^2:S_4$ |