Finite Group DatabasebetaFinite Group Databasebeta| order | 390 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = C4×AGL(1,𝔽13) (order 624) · Inn(G) = C13⋊C3 · Out(G) = C42 |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C2×C5×C13⋊C3 |
| unambiguous semidirect product | (C2×C13)⋊(C3×C5) |
| unambiguous semidirect product | (C2×C5×C13)⋊C3 |
| Fitting, socle, Hall | C2×C5×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C2×C3×C5 |
| Hall | C5×C13 ⋊ C2×C3 |
| Hall | C2×C13 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C5 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 130 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2)(3,4,5,6,7)(9,10,12)(11,14,17)(13,16,15)(18,20,19);(8,9,11,10,13,17,14,18,20,12,15,19,16)70 conjugacy classes — view the full character table.
| maximal subgroup of | C5×Dic13⋊C3, Dic5×C13⋊C3, Dic65⋊C3, … |
| maximal normal subgroup of | C5×Dic13⋊C3, Dic5×C13⋊C3, Dic65⋊C3, … |
| maximal quotient of | C4×C5×C13⋊C3, C22×C5×C13⋊C3, C2×C5×C13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(390, 4);