Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = C22⋊AGL(1,𝔽5) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×Q8 ⋊ C4 |
| Sylow | C5 ⋊ #103 |
42 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)(6,7,9,11)(8,13,12,10);(2,3,4,5)(7,10)(8,12)(11,13)Polycyclic presentation:
2^5,5|a^2=c,b^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^2,d^b=de,f^c=f^419 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×AGL(1,𝔽5), #65735, Q16×AGL(1,𝔽5), … |
| maximal normal subgroup of | SD16×AGL(1,𝔽5), #65735, Q16×AGL(1,𝔽5), … |
| maximal quotient of | #64869, #64872, #64873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 84);