Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = C22×D5 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D4×D5 |
| Fitting | C5×D4 ⋊ C2 |
| Sylow | C5 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D5 |
| commutator, socle, cosocle | C2×C5 | C23 |
| Fitting | C5×D4 | C2 |
| hypercenter | D4 | D5 |
| hypocenter | C5 | C2×D4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ D4 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 3 generators on 9 points:
(2,3)(4,5);(1,2,4,5,3)(7,9);(6,7)(8,9)20 conjugacy classes — view the full character table.
| maximal subgroup of | #4007, D5×D8, #4057, … |
| maximal normal subgroup of | #4007, D5×D8, #4057, … |
| maximal quotient of | #4024, D5×C22⋊C4, #4027, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 39);