Finite Group DatabasebetaFinite Group Databasebeta| order | 280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×C3×C4×S4 (order 576) · Inn(G) = C22 · Out(G) = C2×C3×C4×D3 |
| homology | Abelianization (H1) = C22×C5×C7 · Schur multiplier (H2) = C1 |
| direct product | C5×C7×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C5×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C5×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C7 ◁ G | |
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)(13,14,15,16)(17,20,19,18);(13,17,15,19)(14,18,16,20)175 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(280, 31);