Subgroup ($H$) information
| Description: | $C_3^3:F_9:C_2$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,2,9)(3,4,5)(6,7,8)(10,16,13)(11,17,14)(12,18,15), (1,14,2,12)(3,11,6,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$ |
| Order: | \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$, of order \(559872\)\(\medspace = 2^{8} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:(S_3\times \SD_{16})$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\card{W}$ | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
| Centralizer: | $C_1$ |
| Normalizer: | $C_3^3:F_9:C_2$ |
| Normal closure: | $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $144$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$ |