Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of C5⋊PSU(3,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C5⋊PSU(3,𝔽2) |
| unambiguous semidirect product | (C2×C5)⋊PSU(3,𝔽2) |
| Fitting, socle | C2×C32×C5 ⋊ Q8 |
| Hall | C32×C5 ⋊ C2×Q8 |
| Sylow | C32 ⋊ C2×Dic10 |
| Sylow | C5 ⋊ C2×PSU(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C5⋊PSU(3,𝔽2) |
| commutator, cosocle | C5×C3⋊D3 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | Q8 |
| prime radical | C2×C5×C3⋊D3 | C22 |
| hypocenter | C32×C5 | C2×Q8 |
| LCS | G ▷ C5×C3⋊D3 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
86 conjugacy classes of subgroups.
| transitive degree | μt(G) = 90 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | #13088394, #13088400, #13088482, … |
| maximal normal subgroup of | #13088394, #13088400, #13088482, … |
| maximal quotient of | #13088889, #13088890, C4×C5⋊PSU(3,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 812);