Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×C3×D4×AGL(1,𝔽5) (order 960) · Inn(G) = D5 · Out(G) = C22×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×Dic5 |
| derived, Sylow | C5 ⋊ C2×C4×C7 |
| Hall | C5×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | D5 |
| commutator, hypocenter | C5 | C2×C4×C7 |
| Frattini | C2 | C2×C7×D5 |
| Fitting, socle, prime radical | C22×C5×C7 | C2 |
| cosocle | C2×C5 | C22×C7 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22×C7 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,4,5,6)(8,9)(10,11)(12,13,14,15,16,17,18);(1,2)(7,8,10,11,9)112 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×Dic7, (C2×D7)⋊Dic5, #10587856, … |
| maximal normal subgroup of | Dic5×Dic7, (C2×D7)⋊Dic5, #10587856, … |
| maximal quotient of | C2×C7×#168, C7×#681, C4×C7×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 22);