Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#120 (order 640) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊C4 |
| Fitting | C2×C4×C5 ⋊ C4 |
| Sylow | C5 ⋊ #95 |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,5,4)(6,7,8,9)(11,13);(2,3,4,5)(10,11)(12,13)Polycyclic presentation:
2^5,5|a^2=c,b^2=d,b^a=be,f^a=f^2,f^c=f^428 conjugacy classes — view the full character table.
| maximal subgroup of | #64868, #64869, #64920, … |
| maximal normal subgroup of | #64868, #64869, #64920, … |
| maximal quotient of | #64862, #64863, #64866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 81);