Subgroup ($H$) information
| Description: | $C_3\times D_9^2:S_3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$ac^{16}d^{2}, d^{6}, b^{2}e, c^{14}d^{14}e, c^{9}, b^{3}c^{12}, c^{6}d^{12}, d^{8}, e$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_3\times D_9^2).S_3^2$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^3.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $D_9^2:(C_6\times S_3)$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $W$ | $D_9^2:D_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_3$ |
| Normalizer: | $D_9^2:S_3^2$ |
| Normal closure: | $C_3^5.\SOPlus(4,2)$ |
| Core: | $C_3\times (C_3.(C_3.C_3^3)):C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $(C_3\times D_9^2).S_3^2$ |