Finite Group DatabasebetaFinite Group Databasebeta| order | 1380 |
| isoclinism | isoclinic class of D69 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic69 |
| unambiguous semidirect product | C3⋊(C5×Dic23) |
| unambiguous semidirect product | C23⋊(C5×Dic3) |
| derived, Hall | C3×C23 ⋊ C4×C5 |
| Hall | C3×C5×C23 ⋊ C4 |
| Hall | C5×C23 ⋊ Dic3 |
| Sylow | C23 ⋊ C5×Dic3 |
| Hall | C3×C5 ⋊ Dic23 |
| Sylow | C3 ⋊ C5×Dic23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D69 |
| commutator, hypocenter | C3×C23 | C4×C5 |
| Frattini | C2 | C5×D69 |
| Fitting, socle, prime radical | C2×C3×C5×C23 | C2 |
| cosocle | C2×C3×C23 | C2×C5 |
| LCS | G ▷ C3×C23 | |
| UCS | C1 ◁ C2×C5 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1380 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35);(6,8,7,9)(11,12)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)360 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×Dic207, C3×C5×Dic69, C5×C3⋊Dic69, … |
| maximal normal subgroup of | C3×C5×Dic69, C5×C11×Dic69, C5×C3⋊(Dic23⋊C11), … |
| maximal quotient of | C5×Dic207, C3×C5×Dic69, C5×C3⋊Dic69, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1380, 5);