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) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ C22⋊C8 |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,5,4)(7,9)(10,12,14,13)(11,15,17,16);(2,3,4,5)(6,7)(8,9)(10,11,13,16,14,17,12,15)Polycyclic presentation:
2^5,5|a^2=c,b^2=e,c^2=e,b^a=bd,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 | #64871, #64872, #64887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 78);