Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C23×AGL(1,𝔽5) (order 160) · Inn(G) = D5 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C8×D5 |
| Fitting | C5×C8 ⋊ C2 |
| derived, Sylow | C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D5 |
| commutator, hypocenter | C5 | C2×C8 |
| Frattini | C4 | C2×D5 |
| Fitting | C5×C8 | C2 |
| socle | C2×C5 | C2×C4 |
| prime radical | C2×D5 | C4 |
| cosocle | C4×C5 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C8 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(2,3)(4,5);(1,2,4,5,3)(6,7,8,10,9,11,12,13)32 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C16, #3930, D5⋊C16, … |
| maximal normal subgroup of | D5×C16, #3930, D5⋊C16, … |
| maximal quotient of | D5×C16, #3930, C8×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 4);