Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of D5×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×D5×C22⋊C9 |
| unambiguous semidirect product | C5⋊(C22×C22⋊C9) |
| unambiguous semidirect product | (C2×C5)⋊(C2×C22⋊C9) |
| unambiguous semidirect product | (C23×D5)⋊C9 |
| derived | C22×C5 ⋊ C22×C9 |
| Hall | C23×D5 ⋊ C9 |
| Sylow | C5 ⋊ C22×C22⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5×A4 |
| commutator, hypocenter | C22×C5 | C22×C9 |
| Frattini | C3 | C2×D5×A4 |
| Fitting, socle | C23×C3×C5 | C2×C3 |
| prime radical | C23×C3×D5 | C3 |
| cosocle | C22×C3×C5 | C22×C3 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C2×C3 | |
136 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 90 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C22⋊C9)⋊C4, #13085159, #13085160, … |
| maximal normal subgroup of | D5×(C22⋊C9)⋊C4, #13085159, #13085160, … |
| maximal quotient of | C22⋊C9×Dic10, C4×D5×C22⋊C9, C22⋊C9×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 398);