Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C3×#11925350 |
| Hall | C32×C52 ⋊ C22⋊C4 |
| Sylow | C52 ⋊ C3×#217 |
| Sylow | C32 ⋊ #87831 |
410 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 410 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 120 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3)(5,6)(8,10)(12,13,14,15)(16,17,18,19);(4,5,6)(7,8)(9,10)(11,12,15,13,14)(17,18)(19,20)162 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#72224586, C3×#72224574, C3×D3×#87831, … |
| maximal normal subgroup of | C3×#72224586, C3×#72224574, C3×D3×#87831 |
| maximal quotient of | #72254706 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3)(5,6)(8,10)(12,13,14,15)(16,17,18,19), (4,5,6)(7,8)(9,10)(11,12,15,13,14)(17,18)(19,20));