Subgroup ($H$) information
| Description: | $C_9^4:(C_2\times C_6)$ |
| Order: | \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{6}c^{5}d^{7}e^{6}f^{5}, c^{6}f^{3}, e^{6}, c^{8}f, d^{6}e^{6}, bc^{9}d^{16}e^{10}f^{2}, c^{12}e^{8}f^{3}, f^{4}, d^{2}e^{8}, f^{3}, a^{4}bc^{16}d^{16}e^{6}f^{8}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $D_9\wr C_4.C_3$ |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_9\wr C_4.C_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_9^3.C_6^2.C_3^2.C_6^2.C_2$, of order \(17006112\)\(\medspace = 2^{5} \cdot 3^{12} \) |
| $\card{W}$ | 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 | $4$ |
| Möbius function | not computed |
| Projective image | not computed |