Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = D4×AGL(1,𝔽5) (order 160) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊AGL(1,𝔽5) |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊C2 |
| Fitting | C22×C5 ⋊ C4 |
| Sylow | C5 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C4 |
| Fitting | C22×C5 | C4 |
| prime radical | C22×D5 | C2 |
| cosocle | C2×D5 | C22 |
| hypercenter | C22 | AGL(1,𝔽5) |
| hypocenter | C5 | C22⋊C4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C22 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2,3,5,4)(7,9);(2,3,4,5)(6,7)(8,9)Polycyclic presentation:
2^4,5|a^2=c,b^a=bd,e^a=e^2,e^c=e^414 conjugacy classes — view the full character table.
| maximal subgroup of | #3999, #4011, #4130, … |
| maximal normal subgroup of | #3999, #4011, #4130, … |
| maximal quotient of | #3999, #4003, #4005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 34);