Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C5 |
| direct product | C3×#10597115 |
| Hall | C32×C52 ⋊ D4 |
| Sylow | C52 ⋊ C3×#67 |
| Sylow | C32 ⋊ C4⋊(C5⋊D5) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×C5⋊D5 |
| commutator, cosocle | C2×C3×C52 | C22×C3 |
| Frattini | C2 | C3×D3×C5⋊D5 |
| Fitting, socle | C2×C32×C52 | C22 |
| hypocenter | C3×C52 | C3×D4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2×C3 | |
296 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 300 |
243 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#67×C5⋊D5, #72231388, #72231389, … |
| maximal normal subgroup of | C3×#67×C5⋊D5, #72231388, #72231389, … |
| maximal quotient of | C2×C3×#10597115 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 425);