Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C2×C3×D4×AGL(1,𝔽5) (order 960) · Inn(G) = C2×D5 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×Dic10 |
| unambiguous semidirect product | C5⋊(C7×Q8) |
| Hall | C5×C7 ⋊ Q8 |
| Sylow | C5 ⋊ C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C7 |
| Frattini | C2 | C2×C7×D5 |
| Fitting | C4×C5×C7 | C2 |
| socle, prime radical | C2×C5×C7 | C22 |
| hypercenter | C4×C7 | D5 |
| hypocenter | C5 | C7×Q8 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 | |
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)(8,9,10,11)(12,15,14,13)(16,17,18,19,20);(8,12,10,14)(9,13,11,15)(17,20)(18,19)91 conjugacy classes — view the full character table.
| maximal subgroup of | #10587844, #10587849, #10587850, … |
| maximal normal subgroup of | #10587844, #10587849, #10587850, … |
| maximal quotient of | C7×#683, C7×C4⋊Dic5, C2×C7×Dic10, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 19);