Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽7) (order 672) · Inn(G) = C2×D7 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×C22⋊D7 |
| unambiguous semidirect product | C22⋊(C5×D7) |
| Fitting | C22×C5×C7 ⋊ C2 |
| Hall | C5×C7 ⋊ D4 |
| Sylow | C7 ⋊ C5×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D7 |
| commutator, cosocle | C2×C7 | C22×C5 |
| Frattini | C2 | C2×C5×D7 |
| Fitting | C22×C5×C7 | C2 |
| socle | C2×C5×C7 | C22 |
| hypercenter | C22×C5 | D7 |
| hypocenter | C7 | C5×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
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:
(1,2,4,6,7,5,3)(9,11)(12,13,14,15,16);(2,3)(4,5)(6,7)(8,9)(10,11)85 conjugacy classes — view the full character table.
| maximal subgroup of | #10587952, #10587953, D5×C22⋊D7, … |
| maximal normal subgroup of | #10587952, #10587953, D5×C22⋊D7, … |
| maximal quotient of | C5×Dic7⋊C4, C5×(C2×D7)⋊C4, C5×#1141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 18);