Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#120 (order 640) · Inn(G) = C2×D5 · Out(G) = C2×#41 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×C22⋊D5 |
| unambiguous semidirect product | C23⋊D5 |
| Fitting | C23×C5 ⋊ C2 |
| Sylow | C5 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D5 |
| commutator, cosocle | C2×C5 | C23 |
| Frattini | C2 | C22×D5 |
| Fitting | C23×C5 | C2 |
| socle | C22×C5 | C22 |
| hypercenter | C23 | D5 |
| hypocenter | C5 | C2×D4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C23 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 11 points:
(4,6)(7,8,10,11,9);(3,4)(5,6)(8,9)(10,11);(1,2)26 conjugacy classes — view the full character table.
| maximal subgroup of | #3938, #4011, #4013, … |
| maximal normal subgroup of | #3938, #4011, #4013, … |
| maximal quotient of | C2×#683, #4070, C2×#685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 44);