Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#10597122 |
| Hall | C32×C52 ⋊ D4 |
| Sylow | C52 ⋊ C3×D12 |
| Sylow | C32 ⋊ #6038 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D5×D15 |
| commutator, cosocle | C2×C3×C52 | C22×C3 |
| Frattini | C2 | C3×D5×D15 |
| Fitting, socle | C2×C32×C52 | C22 |
| hypocenter | C3×C52 | C3×D4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2×C3 | |
184 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 184 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.
| transitive degree | μt(G) = 120 |
207 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#11925429, C3×#11925646, C3×D3×#6038, … |
| maximal normal subgroup of | C3×#11925429, C3×#11925646, C3×D3×#6038 |
| maximal quotient of | #72254697 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 432);