Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽7) (order 672) · Inn(G) = D7 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C5×C8 · Schur multiplier (H2) = C1 |
| direct product | C5×C7⋊C8 |
| unambiguous semidirect product | C7⋊(C5×C8) |
| derived, Sylow | C7 ⋊ C5×C8 |
| Hall | C5×C7 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | D7 |
| commutator, hypocenter | C7 | C5×C8 |
| Frattini | C4 | C5×D7 |
| Fitting | C4×C5×C7 | C2 |
| socle, prime radical | C2×C5×C7 | C4 |
| cosocle | C4×C7 | C2×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C4×C5 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3)(4,5)(6,7)(8,13,10,15,11,9,14,12)(16,17,18,19,20);(1,2,4,6,7,5,3)100 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C7⋊C8, #10587837, #10587839, … |
| maximal normal subgroup of | D5×C7⋊C8, #10587837, #10587839, … |
| maximal quotient of | C5×C7⋊C16, C2×C5×C7⋊C8, C3×C5×C7⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 2);