Subgroup ($H$) information
| Description: | $(C_9\times S_3^2):C_6$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$b^{9}, c^{2}d^{2}efg^{2}, b^{6}, cdef^{2}g, b^{14}, g, f, a^{2}e^{2}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.(S_3\times D_4)$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_6.C_6.C_2^3$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_{12}.C_2^3$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $W$ | $S_3^3:C_6$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^6.(S_3\times D_4)$ |