Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C22×C3×AGL(1,𝔽5) (order 240) · Inn(G) = AGL(1,𝔽5) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×#170 |
| derived, Sylow | C5 ⋊ C7×C8 |
| Hall | C5×C7 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C7×C8 |
| Frattini | C2 | C7×AGL(1,𝔽5) |
| Fitting, socle, prime radical | C2×C5×C7 | C4 |
| cosocle | Dic5 | C2×C7 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C7 | |
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:
(2,5,4,3)(6,13,12,11,9,10,8,7)(14,15,16,17,18,19,20);(1,2,3,5,4)70 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#170, #10587920, #10587921, … |
| maximal normal subgroup of | D7×#170, #10587920, #10587921, … |
| maximal quotient of | C7×#674, C2×C7×#170, C3×C7×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 5);