Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#72209641 (order 30720) · Inn(G) = D5 · Out(G) = C2×#72209641 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×C4×D5 |
| Fitting | C22×C4×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | D5 |
| commutator, hypocenter | C5 | C23×C4 |
| Frattini | C2 | C23×D5 |
| Fitting | C22×C4×C5 | C2 |
| socle | C23×C5 | C22 |
| prime radical | C23×D5 | C2 |
| cosocle | C2×C5 | C24 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C22×C4 | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(3,4)(10,11)(12,13);(5,6,7,8)(9,10,12,13,11);(3,4);(1,2)64 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#95, #64953, #64954, … |
| maximal normal subgroup of | D5×#95, #64953, #64954, … |
| maximal quotient of | C2×C4×Dic10, C2×C42×D5, C2×#4018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 214);