Subgroup ($H$) information
| Description: | $\PSU(3,2)$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(243\)\(\medspace = 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,11,6,7)(2,10,5,4)(3,12)(8,9)(13,19,18,17)(14,20,15,16), (1,9,6,5)(2,3) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_3^6.\SL(2,3)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_6.C_6^2.D_6$, of order \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $\PU(3,2)$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $\PU(3,2)$ | ||
| Normal closure: | $C_3^6.Q_8$ | ||
| Core: | $C_3^2$ | ||
| Minimal over-subgroups: | $C_3^4:Q_8$ | $C_3^4:Q_8$ | $\PU(3,2)$ |
| Maximal under-subgroups: | $C_3^2:C_4$ | $Q_8$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $C_3^6.\SL(2,3)$ |