Subgroup ($H$) information
| Description: | $C_3^2\wr C_2.D_4$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(5\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$a^{3}, cef^{2}, de^{2}, b^{5}, b^{10}, f, ef^{2}, a^{2}c^{2}d$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, a Hall subgroup, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^4:C_{20}:C_4$ |
| Order: | \(6480\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:C_{40}:C_4$, of order \(12960\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_4^2.C_2^3$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $W$ | $C_3^2\wr C_2.D_4$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $-1$ |
| Projective image | $C_3^4:C_{20}:C_4$ |