Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = D4×AGL(1,𝔽5) (order 160) · Inn(G) = AGL(1,𝔽5) · Out(G) = D4 |
| holomorph | G ⋊ Aut(G) = #72221744 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×#170 |
| derived, Sylow | C5 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C2×C8 |
| Frattini | C2 | C2×AGL(1,𝔽5) |
| Fitting, socle, prime radical | C22×C5 | C4 |
| cosocle | Dic5 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 15 points:
(4,5,6,7)(8,9,10,12,11,13,14,15);(1,2)(3,4,5,7,6)20 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#170, C4⋊#170, #4002, … |
| maximal normal subgroup of | C4×#170, C4⋊#170, #4002, … |
| maximal quotient of | C2×#674, #3998, C4×#170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 32);