Subgroup ($H$) information
| Description: | $C_6^2.S_3^2$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a^{3}, c^{2}f^{2}, b^{6}, b^{9}d^{2}e^{2}, b^{8}c^{3}d^{4}ef, d^{3}ef^{2}, c^{3}df, a^{2}d^{3}ef^{2}$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^5.(C_6\times S_4)$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.(D_6\times S_4)$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_6^2.S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $W$ | $C_6^2.S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^5.(C_6\times S_4)$ |