Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #3993 |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,ℤ/8) (order 640) · Inn(G) = C4⋊AGL(1,𝔽5) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×C8 ⋊ C4 |
| Sylow | C5 ⋊ #106 |
36 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)(6,7,8,10,9,11,12,13);(2,3,4,5)(7,10)(8,12)(11,13)Polycyclic presentation:
2^5,5|a^2=c,b^2=d,d^2=e,b^a=bd,d^a=de,f^a=f^2,f^c=f^422 conjugacy classes — view the full character table.
| maximal subgroup of | #64847, ΓL(1,𝔽81), #64903, … |
| maximal normal subgroup of | #64847, ΓL(1,𝔽81), #64903, … |
| maximal quotient of | #64847, ΓL(1,𝔽81), #64881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 68);