Finite Group DatabasebetaFinite Group Databasebeta| order | 1620 |
| isoclinism | isoclinic class of D135 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic135 |
| derived | C5×C27 ⋊ C3×C4 |
| Hall | C3×C5×C27 ⋊ C4 |
| Sylow | C3×C27 ⋊ Dic5 |
| Sylow | C5 ⋊ C3×Dic27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D135 |
| commutator, hypocenter | C5×C27 | C3×C4 |
| Frattini | C2×C9 | C3×D15 |
| Fitting | C2×C3×C5×C27 | C2 |
| socle, prime radical | C2×C32×C5 | D9 |
| cosocle | C2×C5×C27 | C2×C3 |
| LCS | G ▷ C5×C27 | |
| UCS | C1 ◁ C2×C3 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 540 |
414 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1620, 39);