Finite Group DatabasebetaFinite Group Databasebeta| order | 2340 |
| isoclinism | isoclinic class of D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C13 · Schur multiplier (H2) = C1 |
| direct product | C5×C13×Dic9 |
| unambiguous semidirect product | C9⋊(C4×C5×C13) |
| derived, Sylow | C9 ⋊ C4×C5×C13 |
| Hall | C5×C9×C13 ⋊ C4 |
| Hall | C9×C13 ⋊ C4×C5 |
| Hall | C5×C9 ⋊ C4×C13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5×C13 | D9 |
| commutator, hypocenter | C9 | C4×C5×C13 |
| Frattini | C2×C3 | C5×D3×C13 |
| Fitting | C2×C5×C9×C13 | C2 |
| socle, prime radical | C2×C3×C5×C13 | D3 |
| cosocle | C2×C9 | C2×C5×C13 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C2×C5×C13 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 2340 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20)(21,22)(23,24,25,26,27,28,29,30,31);(19,21,20,22)(24,31)(25,30)(26,29)(27,28)780 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C13×C25×Dic9, C52×C13×Dic9, C5×C13×Dic45, … |
| maximal normal subgroup of | C13×C25×Dic9, C52×C13×Dic9, C5×C7×C13×Dic9, … |
| maximal quotient of | C13×C25×Dic9, C52×C13×Dic9, C5×C13×Dic45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2340, 9);