Subgroup ($H$) information
| Description: | $C_9:D_9$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{6}c^{6}d^{3}e^{3}f^{3}, e^{3}, c^{8}d^{6}e^{7}f^{8}, d^{3}e^{4}, c^{6}e^{3}f^{6}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_9:D_9^3.C_6$ |
| Order: | \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| 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.
Quotient group ($Q$) structure
| Description: | $D_9^2:C_6$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | $C_9^2:(C_3\times \SD_{16})$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $3$ |
The quotient 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.S_3\wr D_4$, of order \(7558272\)\(\medspace = 2^{7} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.C_3^3:\GL(2,3)$, of order \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |