Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #686 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = C22⋊D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ #108 |
30 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)(6,8,9,17,11,18,10,19)(7,12,13,16,15,20,14,21);(2,3)(4,5)(6,7,10,14,11,15,9,13)(8,16,17,20,18,21,19,12)Polycyclic presentation:
2^5,5|a^2=d,b^2=c,c^2=e,d^2=e,b^a=bd,f^a=f^4,d^b=de34 conjugacy classes — view the full character table.
| maximal subgroup of | #65037, #65038, #65043, … |
| maximal normal subgroup of | #65037, #65038, #65043, … |
| maximal quotient of | #64699, #64700, #64778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 29);