Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×D3×Dic6 |
| unambiguous semidirect product | (C3×D3)⋊(C5×Q8) |
| Hall | C32×C5 ⋊ C2×Q8 |
| Sylow | C32 ⋊ C2×C5×Q8 |
164 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6655, D3×#6660, D3×#6661, … |
| maximal normal subgroup of | D3×#6655, D3×#6660, D3×#6661, … |
| maximal quotient of | C5×Dic3×Dic6, C5×#63845, C5×#63844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 643);