Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×D15 |
| unambiguous semidirect product | C3⋊(D5×SL(2,𝔽3)) |
| unambiguous semidirect product | C5⋊(D3×SL(2,𝔽3)) |
| unambiguous semidirect product | (Q8×D15)⋊C3 |
| Fitting, derived | C3×C5×Q8 ⋊ C2×C3 |
| Sylow | C5 ⋊ D3×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×D15 |
| commutator, Fitting, cosocle, hypocenter | C3×C5×Q8 | C2×C3 |
| socle | C2×C3×C5 | C2×A4 |
| LCS | G ▷ C3×C5×Q8 | |
| UCS | C1 ◁ C2 | |
78 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
63 conjugacy classes — view the full character table.
| maximal subgroup of | D15×#226, #13088551, #13088552, … |
| maximal normal subgroup of | D15×#226, #13088551, #13088552, … |
| maximal quotient of | SL(2,𝔽3)×Dic15, C2×SL(2,𝔽3)×D15 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 545);