Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of #686 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = C22⋊D5 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D5 |
| commutator, cosocle | C4×C5 | C22 |
| Frattini | C4 | C2×D5 |
| Fitting | C5×Q8 | C2 |
| socle, prime radical | C2×C5 | D4 |
| hypercenter | Q8 | D5 |
| hypocenter | C5 | Q16 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,5,3)(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14);(2,3)(4,5)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)Polycyclic presentation:
2^4,5|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^4,c^b=cd17 conjugacy classes — view the full character table.
| maximal subgroup of | #4061, #4062, D5×Q16, … |
| maximal normal subgroup of | #4061, #4062, D5×Q16, … |
| maximal quotient of | #3939, #3942, #3967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 18);