Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic7⋊C9 |
| unambiguous semidirect product | Dic7⋊(C5×C9) |
| unambiguous semidirect product | (C7⋊C9)⋊(C4×C5) |
| unambiguous semidirect product | (C5×Dic7)⋊C9 |
| derived, Sylow | C7 ⋊ C4×C5×C9 |
| Hall | C5×C7⋊C9 ⋊ C4 |
| Hall | C5×Dic7 ⋊ C9 |
| Hall | C7⋊C9 ⋊ C4×C5 |
| Hall | C5×C7 ⋊ C4×C9 |
| Hall | Dic7 ⋊ C5×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C5 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C4×C5×C9 |
| Frattini | C2×C3 | C5×AGL(1,𝔽7) |
| Fitting, socle, prime radical | C2×C3×C5×C7 | C2×C3 |
| cosocle | C2×C3×C7 | C2×C3×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3×C5 | |
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)(7,11,10,8,9,12)(13,20,19,15,14,21,18,17,16)(22,23,24,25);(6,7,10,11,9,12,8)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 1);