Subgroup ($H$) information
| Description: | $C_5:D_5$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Index: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{3}, c^{3}d^{12}, b$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_5\times C_{15}^2):C_6$ |
| Order: | \(6750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{3} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{15}).C_{15}.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_5^2.\GL(2,5)$, of order \(12000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \) |
| $W$ | $C_5:D_5$, of order \(50\)\(\medspace = 2 \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||
| Normalizer: | $C_{15}^2:C_2$ | ||
| Normal closure: | $C_5^3:C_2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_5^3:C_2$ | $C_{15}:D_5$ | $C_{15}:D_5$ |
| Maximal under-subgroups: | $C_5^2$ | $D_5$ | $D_5$ |
Other information
| Number of subgroups in this autjugacy class | $120$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $0$ |
| Projective image | $(C_5\times C_{15}^2):C_6$ |