Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×Dic45 |
| unambiguous semidirect product | C9⋊(C7×Dic5) |
| derived, Hall | C5×C9 ⋊ C4×C7 |
| Hall | C5×C7×C9 ⋊ C4 |
| Hall | C7×C9 ⋊ Dic5 |
| Hall | C5×C7 ⋊ Dic9 |
| Sylow | C9 ⋊ C7×Dic5 |
| Sylow | C5 ⋊ C7×Dic9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D45 |
| commutator, hypocenter | C5×C9 | C4×C7 |
| Frattini | C2×C3 | C7×D15 |
| Fitting | C2×C5×C7×C9 | C2 |
| socle, prime radical | C2×C3×C5×C7 | D3 |
| cosocle | C2×C5×C9 | C2×C7 |
| LCS | G ▷ C5×C9 | |
| UCS | C1 ◁ C2×C7 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1260 |
A minimal faithful permutation representation — 2 generators on 25 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);(8,10,9,11)(13,20)(14,19)(15,18)(16,17)(22,25)(23,24)336 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×C11×Dic45, C7×Dic495, C7×C13×Dic45, … |
| maximal normal subgroup of | C7×C11×Dic45, C7×C13×Dic45, C7×C17×Dic45, … |
| maximal quotient of | C7×C11×Dic45, C7×Dic495, C7×C13×Dic45, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 10);