Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of D35 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C32×Dic35 |
| derived, Hall | C5×C7 ⋊ C32×C4 |
| Hall | C32×C5×C7 ⋊ C4 |
| Hall | C32×C7 ⋊ Dic5 |
| Hall | C32×C5 ⋊ Dic7 |
| Sylow | C7 ⋊ C32×Dic5 |
| Sylow | C5 ⋊ C32×Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D35 |
| commutator, hypocenter | C5×C7 | C32×C4 |
| Frattini | C2 | C32×D35 |
| Fitting, socle, prime radical | C2×C32×C5×C7 | C2 |
| cosocle | C2×C5×C7 | C2×C32 |
| LCS | G ▷ C5×C7 | |
| UCS | C1 ◁ C2×C32 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 1260 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3)(7,8)(9,10)(11,12,13,14,15)(16,17,18,19,20,21,22);(4,5,6)(7,9,8,10)(12,15)(13,14)(17,22)(18,21)(19,20)342 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C32×C11×Dic35, C32×Dic385, C3×#19129703, … |
| maximal normal subgroup of | C32×C11×Dic35, C32×C13×Dic35, C32×C17×Dic35, … |
| maximal quotient of | C32×C11×Dic35, C32×Dic385, C3×#19129703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 51);