Finite Group DatabasebetaFinite Group Databasebeta| order | 2700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C5×C3⋊D3×D15 |
| Fitting, socle, Hall | C33×C52 ⋊ C22 |
| derived | C33×C5 ⋊ C22×C5 |
| Sylow | C33 ⋊ C2×C5×D5 |
| Sylow | C52 ⋊ D3×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊D3×D15 |
| commutator, cosocle, hypocenter | C33×C5 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C33×C52 | C22 |
| LCS | G ▷ C33×C5 | |
| UCS | C1 ◁ C5 | |
368 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3⋊D3×D15⋊D5, C5×D3×D5×C3⋊D3, D5×C3⋊D3×D15 |
| maximal normal subgroup of | C3⋊D3×D15⋊D5, C5×D3×D5×C3⋊D3, D5×C3⋊D3×D15 |
Past the SmallGroups library coverage — no SmallGroups id.