Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×#41⋊C9 |
| unambiguous semidirect product | #41⋊(C5×C9) |
| unambiguous semidirect product | (C5×#41)⋊C9 |
| Hall | C5×#41 ⋊ C9 |
| Sylow | #41 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4×C5 | A4 |
| commutator, hypocenter | Q8 | C2×C5×C9 |
| Frattini | C2×C3 | C2×C5×A4 |
| Fitting, prime radical | C3×C5×#41 | C3 |
| socle | C2×C3×C5 | C2×A4 |
| cosocle | C3×Q8 | C2×C3×C5 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C3×C4×C5 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#63422, #13083944, #4094⋊C9, … |
| maximal normal subgroup of | C5×#63422, #13083944, #4094⋊C9, … |
| maximal quotient of | C4×C5×Q8⋊C9, C2×C5×#41⋊C9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 154);