Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#2021 (order 2560) · Inn(G) = C2×D5 · Out(G) = #2388 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×D5 |
| commutator | C2×C5 | C42 |
| Fitting | C42×C5 | C2 |
| socle, prime radical | C22×C5 | C2×C4 |
| cosocle | C2×C4×C5 | C22 |
| hypercenter | C42 | D5 |
| hypocenter | C5 | #97 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,5,3)(6,7,8,9)(11,14)(15,17);(2,3)(4,5)(10,11,12,15,13,14,16,17)Polycyclic presentation:
2^5,5|a^2=c,b^2=d,c^2=e,b^a=be,f^a=f^452 conjugacy classes — view the full character table.
| maximal subgroup of | #64684, #64685, #64760, … |
| maximal normal subgroup of | #64684, #64685, #64760, … |
| maximal quotient of | #64674, #64686, #64744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 10);