Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×D5×SL(2,𝔽3) |
| unambiguous semidirect product | C5⋊(C2×C3×SL(2,𝔽3)) |
| unambiguous semidirect product | (Q8×D5)⋊C32 |
| unambiguous semidirect product | (C3×Q8×D5)⋊C3 |
| Fitting | C3×C5×Q8 ⋊ C2×C3 |
| derived | C5×Q8 ⋊ C2×C32 |
| Hall | Q8×D5 ⋊ C32 |
| Sylow | C5 ⋊ C2×C3×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5×A4 |
| commutator, cosocle, hypocenter | C5×Q8 | C2×C32 |
| Frattini | C2 | C3×D5×A4 |
| Fitting | C3×C5×Q8 | C2×C3 |
| socle | C2×C3×C5 | C2×A4 |
| LCS | G ▷ C5×Q8 | |
| UCS | C1 ◁ C2×C3 | |
100 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 539);