Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C4×#3025⋊C3 |
| Fitting, Sylow | C4×#3025 ⋊ C3 |
1932 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 1932 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(7,14,8)(10,18,11)(13,16,15)(17,20,19);(5,20)(6,14)(7,11)(8,17)(9,18)(10,15)(12,16)(13,19);(7,13)(8,15)(10,17)(11,19)(14,16)(18,20)128 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#11714162, C4×#11714175, C4×#60399⋊C3, … |
| maximal normal subgroup of | C4×#11714162, C4×#11714175, C4×#60399⋊C3, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(7,14,8)(10,18,11)(13,16,15)(17,20,19), (5,20)(6,14)(7,11)(8,17)(9,18)(10,15)(12,16)(13,19), (7,13)(8,15)(10,17)(11,19)(14,16)(18,20));