Finite Group DatabasebetaFinite Group Databasebeta| order | 810 |
| isoclinism | isoclinic class of #272 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×#272 |
| Fitting, Hall | C3×C5×H(3,𝔽3) ⋊ C2 |
| derived | H(3,𝔽3) ⋊ C2×C3×C5 |
| Sylow | C3×H(3,𝔽3) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C32×C5 | C3⋊D3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C3×C5 |
| Frattini | C3 | C3×C5×C3⋊D3 |
| Fitting | C3×C5×H(3,𝔽3) | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C32×C5 | |
152 conjugacy classes of subgroups.
| transitive degree | μt(G) = 135 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×#1088, C3×#10587382, C5×C3⋊#1088, … |
| maximal normal subgroup of | C3×C5×#1088, C3×#10587382, C5×C3⋊#1088, … |
| maximal quotient of | C3×C5×#1084, C2×C3×C5×#272, C5×#92302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(810, 82);