Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×C5×Dic3 |
| derived | C3 ⋊ C22×C3×C4×C5 |
| Hall | C32×C5 ⋊ C22×C4 |
| Sylow | C32 ⋊ C22×C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3×C5 | D3 |
| commutator, hypocenter | C3 | C22×C3×C4×C5 |
| Frattini | C2 | C22×C3×C5×D3 |
| Fitting, socle, prime radical | C23×C32×C5 | C2 |
| cosocle | C2×C3 | C23×C3×C5 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23×C3×C5 | |
248 conjugacy classes of subgroups.
| transitive degree | μt(G) = 240 |
360 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×#90994, C5×#63579, #13084870, … |
| maximal normal subgroup of | C3×#90994, C5×#63579, #13084870, … |
| maximal quotient of | C22×C3×C5×C3⋊C8, C2×C3×C5×#208, C2×C3×C4×C5×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 727);