Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(3,𝔽2) (order 26880) · Inn(G) = D5 · Out(G) = C2×AGL(3,𝔽2) |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C23×D5 |
| unambiguous semidirect product | C5⋊C24 |
| unambiguous semidirect product | (C2×C5)⋊C23 |
| Fitting, socle | C23×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D5 |
| commutator, cosocle, hypocenter | C5 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C23×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C23 | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 4 generators on 11 points:
(1,2)(3,4)(5,6)(8,9)(10,11);(5,6)(7,8,10,11,9);(3,4);(1,2)32 conjugacy classes — view the full character table.
| maximal subgroup of | D5×C22⋊C4, #4028, C2×#685, … |
| maximal normal subgroup of | D5×C22⋊C4, #4028, C2×#685, … |
| maximal quotient of | C22×Dic10, C22×C4×D5, C22×D20, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 51);