Finite Group DatabasebetaFinite Group Databasebeta| order | 5184 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | C3×C9⋊#5935 |
| Fitting | C3×C9×H(5,𝔽2) ⋊ D3 |
| derived | C9×#997 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C9⋊#1020 |
| commutator, cosocle, hypocenter | C9×#997 | C2×C3 |
| Frattini | C2×C3 | C3×C3⋊#1020 |
| Fitting | C3×C9×H(5,𝔽2) | D3 |
| socle | C2×C32 | C3⋊#1020 |
| LCS | G ▷ C9×#997 | |
| UCS | C1 ◁ C2×C3 | |
825 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 825 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3)(4,10,9,8,7,5,11,6)(13,15)(14,18)(16,19)(17,20);(12,17,15,14,19,16,18,13,20);(5,9,6)(8,10,11)171 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(4,10,9,8,7,5,11,6)(13,15)(14,18)(16,19)(17,20), (12,17,15,14,19,16,18,13,20), (5,9,6)(8,10,11));