Finite Group DatabasebetaFinite Group Databasebeta| order | 390 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C3×C42×D3 (order 288) · Inn(G) = D3 · Out(G) = C3×C42 |
| homology | Abelianization (H1) = C2×C5×C13 · Schur multiplier (H2) = C1 |
| direct product | C5×D3×C13 |
| unambiguous semidirect product | C3⋊(C2×C5×C13) |
| Fitting, socle, Hall | C3×C5×C13 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C5×C13 |
| Hall | C3×C13 ⋊ C2×C5 |
| Hall | C3×C5 ⋊ C2×C13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5×C13 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C5×C13 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C13 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C5×C13 | |
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)(9,10,11,12,13,14,15,16,17,18,19,20,21);(1,2,3)195 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D3×D13, D3×D5×C13, D3×D65, … |
| maximal normal subgroup of | C5×D3×D13, D3×D5×C13, D3×D65, … |
| maximal quotient of | C5×Dic3×C13, C2×C5×D3×C13, C5×C13×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(390, 8);