Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#511790 (order 10240) · Inn(G) = C22×D5 · Out(G) = #36614 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, prime radical, cosocle | C22×C5 | C23 |
| Fitting | C42×C5 | C2 |
| hypercenter | C42 | D5 |
| hypocenter | C5 | C4⋊Q8 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C42 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,4,5,3)(6,7,8,9);(10,12,13,17)(11,14,15,16);(2,3)(4,5)(7,9)(10,11,13,15)(12,16,17,14)Polycyclic presentation:
2^5,5|a^2=e,b^2=e,c^2=d,b^a=be,c^a=cd,f^a=f^446 conjugacy classes — view the full character table.
| maximal subgroup of | #64701, #64702, #64706, … |
| maximal normal subgroup of | #64701, #64702, #64706, … |
| maximal quotient of | #64945, #64950, #64969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 90);