Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = D7 · Out(G) = C3×C4×D4 |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C2×C5×Dic7 |
| unambiguous semidirect product | (C2×C7)⋊(C4×C5) |
| derived, Sylow | C7 ⋊ C2×C4×C5 |
| Hall | C5×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | D7 |
| commutator, hypocenter | C7 | C2×C4×C5 |
| Frattini | C2 | C2×C5×D7 |
| Fitting, socle, prime radical | C22×C5×C7 | C2 |
| cosocle | C2×C7 | C22×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C22×C5 | |
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)(7,8,9,10,11)(13,14)(15,16)(17,18);(1,2)(12,13,15,17,18,16,14)100 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×Dic7, #10587855, #10587856, … |
| maximal normal subgroup of | Dic5×Dic7, #10587855, #10587856, … |
| maximal quotient of | C2×C5×C7⋊C8, C5×#1136, C4×C5×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 17);