Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C32×AΓL(1,𝔽9) |
| Fitting, socle | C2×C34 ⋊ SD16 |
| Sylow | C34 ⋊ C2×SD16 |
590 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 590 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 17 points:
(4,5,6)(7,12,11,14)(8,10,9,15);(1,2,3)(16,17);(8,10)(9,11)(13,14)162 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C32×AGL(2,𝔽3), C2×C3⋊D3×AΓL(1,𝔽9), C22×C32×AΓL(1,𝔽9), … |
| maximal normal subgroup of | C2×C3⋊D3×AΓL(1,𝔽9), C22×C32×AΓL(1,𝔽9), C2×C3×D3×AΓL(1,𝔽9), … |
| maximal quotient 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)(7,12,11,14)(8,10,9,15), (1,2,3)(16,17), (8,10)(9,11)(13,14));