Finite Group DatabasebetaFinite Group Databasebeta| order | 1260 |
| isoclinism | isoclinic class of #4380 |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽5)×AGL(1,𝔽9) (order 8640) · Inn(G) = #4380 · Out(G) = C2×C3×C8 |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C3 |
| direct product | C7×#4380 |
| Fitting, socle, Hall | C32×C5×C7 ⋊ C4 |
| derived, Hall | C32×C5 ⋊ C4×C7 |
| Hall | C32×C7 ⋊ AGL(1,𝔽5) |
| Hall | C5×C7 ⋊ #157 |
| Sylow | C32 ⋊ C7×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C7×#157 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #4380 |
| commutator, hypocenter | C32×C5 | C4×C7 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C7 | C4 |
| prime radical | C7×C3⋊D15 | C2 |
| cosocle | C3⋊D15 | C2×C7 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C7 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 210 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7)(8,9)(10,12,13,11)(15,16,17,18);(8,10,13)(9,12,11)(14,15,16,18,17)105 conjugacy classes — view the full character table.
| maximal subgroup of | C7×C11×#4380, C7×C11⋊#4380, C7×C13×#4380, … |
| maximal normal subgroup of | C7×C11×#4380, C7×C13×#4380, C7×C17×#4380, … |
| maximal quotient of | C7×C11×#4380, C7×C11⋊#4380, C7×C13×#4380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1260, 90);