Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D3×D15 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×D3×D15 |
| Fitting, socle | C22×C32×C5 ⋊ C22 |
| derived, Hall | C32×C5 ⋊ C24 |
| Sylow | C32 ⋊ C23×D5 |
| Sylow | C5 ⋊ C22×D32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D15 |
| commutator, cosocle, hypocenter | C32×C5 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C5 | C22 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
568 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6667, #13085860, #13085865, … |
| maximal normal subgroup of | D3×#6667, #13085860, #13085865, … |
| maximal quotient of | C2×Dic6×D15, C2×#10624193, C2×#10624194, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 831);