Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #3958 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#442 (order 1280) · Inn(G) = #686 · Out(G) = C42 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C5×D8 ⋊ C2 |
| Sylow | C5 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #686 |
| commutator, cosocle | C5×C8 | C22 |
| Frattini | C8 | C2×D5 |
| Fitting | C5×D8 | C2 |
| socle | C2×C5 | D8 |
| hypercenter | D8 | D5 |
| hypocenter | C5 | D16 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
28 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)(7,11)(8,15)(9,16)(12,13)(14,20)(17,21)(18,19);(2,3)(4,5)(6,7)(8,11)(9,18)(10,14)(12,15)(13,16)(17,20)(19,21)Polycyclic presentation:
2^5,5|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,c^b=cd,d^b=de25 conjugacy classes — view the full character table.
| maximal subgroup of | D5×D16, #65200, #65203, … |
| maximal normal subgroup of | D5×D16, #65200, #65203, … |
| maximal quotient of | #64709, #64711, #64739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 33);