Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×C4×AGL(1,𝔽5) (order 160) · Inn(G) = D5 · Out(G) = C22×C4 |
| holomorph | G ⋊ Aut(G) = #72220816 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D5 |
| commutator, hypocenter | C5 | C16 |
| Fitting, cosocle | C5×C8 | C2 |
| socle, prime radical | C2×C5 | C8 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C8 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3)(4,5)(6,7,8,11,9,12,14,17,10,13,15,18,16,19,20,21);(1,2,4,5,3)Polycyclic presentation:
2^4,5|a^2=b,b^2=c,c^2=d,e^a=e^432 conjugacy classes — view the full character table.
| maximal subgroup of | #3928, D5×C16, #3930, … |
| maximal normal subgroup of | #3928, D5×C16, #3930, … |
| maximal quotient of | #3926, C2×#672, C3×#672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 1);