Subgroup ($H$) information
| Description: | $S_3^2:D_9^2$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a^{3}e^{6}f^{6}g^{3}, e^{3}g^{3}, d^{6}ef^{3}g^{7}, d^{6}f^{4}g, f^{3}g^{3}, b^{2}cd^{14}e^{4}f^{4}g^{6}, c^{2}d^{11}e^{4}g^{7}, g^{3}, d^{6}, c^{2}d^{14}e^{6}f^{6}g^{6}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $D_9^2\wr C_2.C_6$ |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_9\wr A_4.C_6$, of order \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^3.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $18$ |
| Möbius function | not computed |
| Projective image | not computed |