Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C27 · Schur multiplier (H2) = C3 |
| direct product | C3×D5×C27 |
| unambiguous semidirect product | C5⋊(C2×C3×C27) |
| Fitting, Hall | C3×C5×C27 ⋊ C2 |
| derived, Sylow | C5 ⋊ C2×C3×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C27 | D5 |
| commutator, hypocenter | C5 | C2×C3×C27 |
| Frattini | C9 | C32×D5 |
| Fitting | C3×C5×C27 | C2 |
| socle | C32×C5 | C2×C9 |
| prime radical | C32×D5 | C9 |
| cosocle | C5×C9 | C2×C32 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C3×C27 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 405 |
324 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×AGL(1,𝔽5)×C27, C3×C27⋊AGL(1,𝔽5), C27×ΓL(1,𝔽16), … |
| maximal normal subgroup of | C3×AGL(1,𝔽5)×C27, C3×C27⋊AGL(1,𝔽5), C27×ΓL(1,𝔽16), … |
| maximal quotient of | C3×Dic5×C27, C2×C3×D5×C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 8);