Finite Group DatabasebetaFinite Group Databasebeta| order | 1584 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×AΓL(1,𝔽9) |
| unambiguous semidirect product | AGL(1,𝔽9)⋊(C2×C11) |
| unambiguous semidirect product | PSU(3,𝔽2)⋊(C2×C11) |
| Fitting, socle, Hall | C32×C11 ⋊ SD16 |
| Sylow | C32 ⋊ C11×SD16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C11 | AΓL(1,𝔽9) |
| commutator, cosocle | #157 | C22×C11 |
| Frattini | C1 | G |
| Fitting, socle | C32×C11 | SD16 |
| prime radical | C11×GO+(4,𝔽2) | C2 |
| hypocenter | C32 | C11×SD16 |
| LCS | G ▷ #157 ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C11 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 99 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,17,19)(13,16,18,15,20,14);(13,14,15,19,16,18,20,17)99 conjugacy classes — view the full character table.
| maximal subgroup of | C11×AGL(2,𝔽3), C11⋊C5×AΓL(1,𝔽9), D11×AΓL(1,𝔽9) |
| maximal normal subgroup of | C11⋊C5×AΓL(1,𝔽9), D11×AΓL(1,𝔽9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1584, 653);