Subgroup ($H$) information
| Description: | $C_5\times C_3^3:A_4$ |
| Order: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$a^{2}, d, b^{5}c^{15}de^{10}, c^{15}d, b^{2}c^{24}e^{12}, e^{10}, c^{20}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_5^3:S_3\wr C_3$ |
| Order: | \(81000\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_{10}^2.C_5.C_{12}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_3\wr S_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $W$ | $S_3\wr C_3$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $25$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_5^3:S_3\wr C_3$ |