Subgroup ($H$) information
| Description: | $(C_3^2\times C_6).S_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(125\)\(\medspace = 5^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a, d^{15}e^{9}f^{11}, d^{20}, a^{2}, cd^{22}f^{11}, f^{10}, e^{10}, bcd^{8}e^{14}f^{14}$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, a Hall subgroup, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_{15}^3.A_4):C_4$ |
| Order: | \(162000\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5^{3} \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{15}\wr S_3.C_4$, of order \(648000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_3^3:D_6$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $W$ | $C_3^3:S_4$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $125$ |
| Möbius function | $-1$ |
| Projective image | $(C_{15}^3.A_4):C_4$ |