Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D3×ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D3×ΓL(1,𝔽16) |
| unambiguous semidirect product | (C3×D3)⋊(C5×AGL(1,𝔽5)) |
| Fitting, socle, Hall | C32×C52 ⋊ C2×C4 |
| derived | C32×C5 ⋊ C2×C4×C5 |
| Sylow | C52 ⋊ D3×Dic3 |
| Sylow | C32 ⋊ C2×C5×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D3×ΓL(1,𝔽16) |
| commutator, hypocenter | C32×C5 | C2×C4×C5 |
| Frattini | C1 | G |
| Fitting, socle | C32×C52 | C2×C4 |
| prime radical | C3×C5×D3×D5 | C2 |
| cosocle | C32×D5 | C22×C5 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C5 | |
149 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×ΓL(1,𝔽16), C5×D32×AGL(1,𝔽5), D3×ΓL(1,𝔽16)⋊D5, … |
| maximal normal subgroup of | D3×D5×ΓL(1,𝔽16), C5×D32×AGL(1,𝔽5), D3×ΓL(1,𝔽16)⋊D5 |
| maximal quotient of | D3×C3⋊AGL(1,ℤ/25) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 625);