Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C2×C36 |
| direct product | C3×A4×AΣL(1,𝔽27) |
| Fitting | C22×C3×AΣL(1,𝔽27) ⋊ C3 |
| Sylow | C22 ⋊ C32×AΣL(1,𝔽27) |
2798 conjugacy classes of subgroups; only the 292 normal subgroups are shown.
This group has 2798 conjugacy classes of subgroups — too many to draw. Only its 292 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,4)(2,3)(5,6,7)(9,13,11)(12,15,16);(8,9,12,10,13,16,14,11,15);(2,4,3);(9,11,13)204 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#10624749, C3×A4×#4183, C3×S4×AΣL(1,𝔽27), … |
| maximal normal subgroup of | A4×#10624749, C3×A4×#4183, C3×S4×AΣL(1,𝔽27), … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4)(2,3)(5,6,7)(9,13,11)(12,15,16), (8,9,12,10,13,16,14,11,15), (2,4,3), (9,11,13));