Finite Group DatabasebetaFinite Group Databasebeta| order | 390 |
| isoclinism | isoclinic class of D13 |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽13) (order 1248) · Inn(G) = D13 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C3×C5×D13 |
| Fitting, socle, Hall | C3×C5×C13 ⋊ C2 |
| derived, Sylow | C13 ⋊ C2×C3×C5 |
| Hall | C5×C13 ⋊ C2×C3 |
| Hall | C3×C13 ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | D13 |
| commutator, cosocle, hypocenter | C13 | C2×C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C13 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C3×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:
(1,2,3)(4,5,6,7,8)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21);(9,10,12,14,16,18,20,21,19,17,15,13,11)120 conjugacy classes — view the full character table.
| maximal subgroup of | C3×C5×#261, C3×#63118, C5×C3⋊#261, … |
| maximal normal subgroup of | C3×C5×#261, C3×#63118, C5×C3⋊#261, … |
| maximal quotient of | C3×C5×Dic13, C2×C3×C5×D13, C5×C9×D13, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(390, 5);