Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| isoclinism | isoclinic class of AGL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C8 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C32×AGL(1,𝔽9) |
| Fitting, socle | C22×C34 ⋊ C8 |
| derived | C32 ⋊ C22×C32×C8 |
| Sylow | C34 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | AGL(1,𝔽9) |
| commutator, hypocenter | C32 | C22×C32×C8 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C8 |
| prime radical | C22×C32×C3⋊D3 | C4 |
| cosocle | #157 | C23×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
662 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 3 generators on 19 points:
(4,5,6)(7,10,15)(8,13,9)(11,14,12)(18,19);(1,2,3)(8,9,10,13,11,14,15,12);(16,17)324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C32×AΓL(1,𝔽9) |
| maximal normal subgroup of | C22×C32×AΓL(1,𝔽9) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,5,6)(7,10,15)(8,13,9)(11,14,12)(18,19), (1,2,3)(8,9,10,13,11,14,15,12), (16,17));