Finite Group DatabasebetaFinite Group Databasebeta| order | 390 |
| isoclinism | isoclinic class of D39 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×D39 |
| unambiguous semidirect product | C3⋊(C5×D13) |
| unambiguous semidirect product | C13⋊(C5×D3) |
| Fitting, socle, Hall | C3×C5×C13 ⋊ C2 |
| derived, Hall | C3×C13 ⋊ C2×C5 |
| Hall | C5×C13 ⋊ D3 |
| Hall | C3×C5 ⋊ D13 |
| Sylow | C13 ⋊ C5×D3 |
| Sylow | C3 ⋊ C5×D13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D39 |
| commutator, cosocle, hypocenter | C3×C13 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C13 | C2 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C5 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 195 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3)(4,5,6,7,8)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21);(1,2,3)(9,10,12,14,16,18,20,21,19,17,15,13,11)105 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×D13, D5×D39, D15⋊D13, … |
| maximal normal subgroup of | C5×D3×D13, D5×D39, D15⋊D13, … |
| maximal quotient of | C5×Dic39, C2×C5×D39, C5×D117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(390, 9);