Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×C4×AGL(1,𝔽7) (order 672) · Inn(G) = D7 · Out(G) = C22×C3×C4 |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C4×C5×D7 |
| Fitting | C4×C5×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C4×C5 |
| Hall | C5×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | D7 |
| commutator, hypocenter | C7 | C2×C4×C5 |
| Frattini | C2 | C2×C5×D7 |
| Fitting | C4×C5×C7 | C2 |
| socle | C2×C5×C7 | C22 |
| prime radical | C2×C5×D7 | C2 |
| cosocle | C2×C7 | C22×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C4×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:
(2,3)(4,5)(6,7)(8,9,10,11,12);(1,2,4,6,7,5,3)(13,14,15,16)100 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#168, #10587838, C5×C8×D7, … |
| maximal normal subgroup of | D7×#168, #10587838, C5×C8×D7, … |
| maximal quotient of | C5×C8×D7, C5×#1131, C4×C5×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 15);