Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C4×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C2×D7 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic14 |
| unambiguous semidirect product | C7⋊(C5×Q8) |
| Hall | C5×C7 ⋊ Q8 |
| Sylow | C7 ⋊ C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D7 |
| commutator, cosocle | C2×C7 | C22×C5 |
| Frattini | C2 | C2×C5×D7 |
| Fitting | C4×C5×C7 | C2 |
| socle, prime radical | C2×C5×C7 | C22 |
| hypercenter | C4×C5 | D7 |
| hypocenter | C7 | C5×Q8 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 280 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5)(6,7,8,9)(10,13,12,11)(14,15,16,17,18,19,20);(6,10,8,12)(7,11,9,13)(15,20)(16,19)(17,18)85 conjugacy classes — view the full character table.
| maximal subgroup of | #10587845, #10587847, #10587850, … |
| maximal normal subgroup of | #10587845, #10587847, #10587850, … |
| maximal quotient of | C5×Dic7⋊C4, C5×C4⋊Dic7, C2×C5×Dic14, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 14);