Finite Group DatabasebetaFinite Group Databasebeta| order | 2100 |
| isoclinism | isoclinic class of D525 |
| automorphism | Aut(G) order 252000 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D1050 |
| direct product | C2×D525 |
| unambiguous semidirect product | (C2×C3)⋊D175 |
| unambiguous semidirect product | (C2×C7)⋊D75 |
| unambiguous semidirect product | (C2×C25)⋊D21 |
| Fitting | C2×C3×C7×C25 ⋊ C2 |
| derived, Hall | C3×C7×C25 ⋊ C22 |
| Hall | C7×C25 ⋊ C2×D3 |
| Hall | C3×C25 ⋊ C2×D7 |
| Sylow | C25 ⋊ C2×D21 |
| Hall | C3×C7 ⋊ C2×D25 |
| Sylow | C7 ⋊ C2×D75 |
| Sylow | C3 ⋊ C2×D175 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D525 |
| commutator, cosocle, hypocenter | C3×C7×C25 | C22 |
| Frattini | C5 | C2×D105 |
| Fitting | C2×C3×C7×C25 | C2 |
| socle | C2×C3×C5×C7 | D5 |
| LCS | G ▷ C3×C7×C25 | |
| UCS | C1 ◁ C2 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1050 |
A minimal faithful permutation representation — 2 generators on 37 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);(4,5)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(32,37)(33,36)(34,35)528 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×D3675, D7×D525, C2×C7×D525, … |
| maximal normal subgroup of | C2×C7×D525, C2×C11×D525, C2×C13×D525, … |
| maximal quotient of | C2×D3675, D7×D525, C2×C7×D525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2100, 40);