Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C32 |
| direct product | C3×Dic3×PSU(3,𝔽2)⋊C3 |
| derived | C3×PSU(3,𝔽2) ⋊ C32×C4 |
830 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 830 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 96 |
A minimal faithful permutation representation — 2 generators on 19 points:
(5,6)(7,8,10)(9,14,15)(11,13,12)(16,17,18,19);(1,2,3)(4,5,6)(8,14,12,9)(10,11,13,15)180 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#67×PSU(3,𝔽2)⋊C3, #626×PSU(3,𝔽2)⋊C3, #624×PSU(3,𝔽2)⋊C3, … |
| maximal normal subgroup of | C3×#67×PSU(3,𝔽2)⋊C3, #626×PSU(3,𝔽2)⋊C3, #624×PSU(3,𝔽2)⋊C3, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,10)(9,14,15)(11,13,12)(16,17,18,19), (1,2,3)(4,5,6)(8,14,12,9)(10,11,13,15));