Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| direct product | D3×SL(2,𝔽5) |
| unambiguous semidirect product | C3⋊(C2×SL(2,𝔽5)) |
| derived | C3×SL(2,𝔽5) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×A5 |
| commutator, hypocenter | C3×SL(2,𝔽5) | C2 |
| Fitting, socle, cosocle | C2×C3 | C2×A5 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
66 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
27 conjugacy classes — view the full character table.
| maximal subgroup of | #13088289, D3×#6729, #13088292, … |
| maximal normal subgroup of | #13088289, D3×#6729, #13088292, … |
| maximal quotient of | Dic3×SL(2,𝔽5), C2×D3×SL(2,𝔽5), S4×SL(2,𝔽5), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 417);