Subgroup ($H$) information
| Description: | $D_5^3$ |
| Order: | \(1000\)\(\medspace = 2^{3} \cdot 5^{3} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\langle(8,12)(10,15)(17,18), (2,14,5,11,7)(4,12,15,10,8), (2,5,7,14,11), (5,11)(7,14)(8,12)(10,15), (1,3)(6,9)(8,12)(10,15), (1,9,13,6,3)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_5^3:(S_3\times S_4)$ |
| Order: | \(18000\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times D_5^3.D_6$, of order \(72000\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $F_5\wr S_3$, of order \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $W$ | $D_5\wr S_3$, of order \(6000\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-3$ |
| Projective image | $C_5^3:(S_3\times S_4)$ |