Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of #677 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = D20 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×C8 ⋊ C2 |
| Sylow | C5 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D20 |
| commutator, cosocle | C4×C5 | C22 |
| Frattini | C4 | C2×D5 |
| Fitting | C5×C8 | C2 |
| socle | C2×C5 | D4 |
| prime radical | D20 | C2 |
| hypercenter | C8 | D5 |
| hypocenter | C5 | SD16 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,4,5,3)(6,7,8,10,9,11,12,13);(2,3)(4,5)(7,10)(8,12)(11,13)Polycyclic presentation:
2^4,5|b^2=c,c^2=d,b^a=bc,c^a=cd,e^a=e^423 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#677, #4050, #4054, … |
| maximal normal subgroup of | C2×#677, #4050, #4054, … |
| maximal quotient of | #3948, #3949, #3953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 6);