Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #625 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×(C2×D3)⋊Dic3 |
| unambiguous semidirect product | (C2×D3)⋊(C5×Dic3) |
| Hall | C32×C5 ⋊ C22⋊C4 |
| Sylow | C32 ⋊ C5×C22⋊C4 |
152 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 152 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 240 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5)(6,7,8,9)(11,13)(15,16)(17,18,19);(10,11)(12,13)(14,15,16)(18,19)150 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C2×D3)⋊Dic3, #13085856, #13085860, … |
| maximal normal subgroup of | D5×(C2×D3)⋊Dic3, #13085856, #13085860, … |
| maximal quotient of | C5×(C2×D3)⋊(C3⋊C8), C5×#63558, C5×#63560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 257);