Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of ΓL(2,𝔽4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×ΓL(2,𝔽4) |
| unambiguous semidirect product | C3⋊(C5×S5) |
| unambiguous semidirect product | A5⋊(C5×D3) |
| Fitting | C3×C5 ⋊ S5 |
| socle | C3×C5×A5 ⋊ C2 |
| derived | C3×A5 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | ΓL(2,𝔽4) |
| commutator, cosocle, hypocenter | C3×A5 | C2×C5 |
| Frattini | C1 | G |
| Fitting | C3×C5 | S5 |
| socle | C3×C5×A5 | C2 |
| LCS | G ▷ C3×A5 | |
| UCS | C1 ◁ C5 | |
90 conjugacy classes of subgroups.
| transitive degree | μt(G) = 75 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 556);