Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C23×C3×AGL(1,𝔽5) (order 480) · Inn(G) = C2×D5 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×C22⋊D5 |
| unambiguous semidirect product | C22⋊(C7×D5) |
| Fitting | C22×C5×C7 ⋊ C2 |
| Hall | C5×C7 ⋊ D4 |
| Sylow | C5 ⋊ C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C7 |
| Frattini | C2 | C2×C7×D5 |
| Fitting | C22×C5×C7 | C2 |
| socle | C2×C5×C7 | C22 |
| hypercenter | C22×C7 | D5 |
| hypocenter | C5 | C7×D4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C7 ◁ C22×C7 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 140 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,5,3)(7,9)(10,11,12,13,14,15,16);(2,3)(4,5)(6,7)(8,9)91 conjugacy classes — view the full character table.
| maximal subgroup of | #10587953, #10587955, D7×C22⋊D5, … |
| maximal normal subgroup of | #10587953, #10587955, D7×C22⋊D5, … |
| maximal quotient of | C7×#683, C7×#685, C7×#686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 23);