Subgroup ($H$) information
| Description: | $C_3^2.A_4$ |
| Order: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Index: | \(1500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$b^{2}cd^{13}e^{4}f^{13}, d^{15}f^{3}, f^{10}, c^{2}f^{5}, c^{3}d^{15}f^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^3:D_5\wr S_3$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^3.C_6^2.(C_{12}\times S_3^2)$ |
| $\operatorname{Aut}(H)$ | $\He_3:S_4$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| $W$ | $C_3^2:S_4$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $250$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^3:D_5\wr S_3$ |