Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = AGL(1,𝔽5)×H(5,𝔽2) (order 640) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D5⋊M16 |
| Sylow | C5 ⋊ C2×M16 |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,3,5,4)(7,12)(11,13);(6,8,10,9)(7,11,12,13);(2,3,4,5)(6,7,9,13,10,12,8,11)Polycyclic presentation:
2^5,5|a^2=d,b^2=e,d^2=e,c^a=ce,f^a=f^2,f^d=f^428 conjugacy classes — view the full character table.
| maximal subgroup of | #64862, #64863, #64894, … |
| maximal normal subgroup of | #64862, #64863, #64894, … |
| maximal quotient of | #65676, C4×#700, #65678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 202);