Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D15×ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | D15×ΓL(1,𝔽16) |
| Fitting, socle, derived, Hall | C32×C52 ⋊ C2×C4 |
| Sylow | C52 ⋊ D3×Dic3 |
| Sylow | C32 ⋊ D5×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C52 | C2×C4 |
| prime radical | C3×D5×D15 | C2 |
| cosocle | C32×C5×D5 | C22 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 | |
149 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 38 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 |
81 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D5×ΓL(1,𝔽16), D3×AGL(1,𝔽5)×D15, #72231129 |
| maximal normal subgroup of | D3×D5×ΓL(1,𝔽16), D3×AGL(1,𝔽5)×D15, #72231129 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 636);