Finite Group DatabasebetaFinite Group Databasebeta| order | 1740 |
| isoclinism | isoclinic class of D435 |
| automorphism | Aut(G) order 194880 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D870 |
| direct product | C2×D435 |
| unambiguous semidirect product | (C2×C3)⋊D145 |
| unambiguous semidirect product | (C2×C5)⋊D87 |
| unambiguous semidirect product | (C2×C29)⋊D15 |
| Fitting, socle | C2×C3×C5×C29 ⋊ C2 |
| derived, Hall | C3×C5×C29 ⋊ C22 |
| Hall | C5×C29 ⋊ C2×D3 |
| Hall | C3×C29 ⋊ C2×D5 |
| Sylow | C29 ⋊ C2×D15 |
| Hall | C3×C5 ⋊ C2×D29 |
| Sylow | C5 ⋊ C2×D87 |
| Sylow | C3 ⋊ C2×D145 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D435 |
| commutator, cosocle, hypocenter | C3×C5×C29 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C29 | C2 |
| LCS | G ▷ C3×C5×C29 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 870 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(4,5)(7,10)(8,9)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)438 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D11×D435, C2×D4785, C2×C11×D435, … |
| maximal normal subgroup of | C2×C11×D435, C2×C13×D435, C2×C17×D435, … |
| maximal quotient of | D11×D435, C2×D4785, C2×C11×D435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1740, 36);