Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#120 (order 640) · Inn(G) = D5 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| direct product | C4×Dic5 |
| derived, Sylow | C5 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D5 |
| commutator, hypocenter | C5 | C42 |
| Frattini | C22 | C2×D5 |
| Fitting | C2×C4×C5 | C2 |
| socle, prime radical, cosocle | C22×C5 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2×C4 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3)(4,5)(10,11,12,13);(1,2,4,5,3)(6,7,8,9)32 conjugacy classes — view the full character table.
| maximal subgroup of | C8×Dic5, #3946, #3947, … |
| maximal normal subgroup of | C8×Dic5, #3946, #3947, … |
| maximal quotient of | C4×#168, #3935, C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 11);