Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#13103996 |
| Sylow | C52 ⋊ C2×#390 |
3138 conjugacy classes of subgroups; only the 339 normal subgroups are shown.
This group has 3138 conjugacy classes of subgroups — too many to draw. Only its 339 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) = 160 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,7,10,9)(14,16)(18,20);(11,12);(2,3,4,5)(7,9,10,8)(14,16)(17,18)(19,20);(2,3,4,5)(7,8,10,9)(13,14,15,16)128 conjugacy classes — view the full character table.
| maximal quotient of | #72226602 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,4,3)(6,7,10,9)(14,16)(18,20), (11,12), (2,3,4,5)(7,9,10,8)(14,16)(17,18)(19,20), (2,3,4,5)(7,8,10,9)(13,14,15,16));