Subgroup ($H$) information
| Description: | $\SOPlus(4,2)^2.C_2$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(3,6,4)(7,17,8)(11,12,18), (4,8,6,18)(7,11,12,17), (2,14)(4,6)(5,13)(7,12) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_3^3\times C_6).\GL(2,3)\wr C_2$ |
| Order: | \(746496\)\(\medspace = 2^{10} \cdot 3^{6} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $6$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_3^4.Q_8^2.S_3\wr C_2$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_4^2.C_2^4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $\card{W}$ | 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 |