Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#87524 (order 7680) · Inn(G) = D5 · Out(G) = C2×#87524 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C42×D5 |
| Fitting | C42×C5 ⋊ C2 |
| derived, Sylow | C5 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | D5 |
| commutator, hypocenter | C5 | C2×C42 |
| Frattini | C22 | C22×D5 |
| Fitting | C42×C5 | C2 |
| socle, cosocle | C22×C5 | C23 |
| prime radical | C22×D5 | C22 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C42 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,3)(4,5);(1,2,4,5,3)(10,11,12,13);(6,7,8,9)64 conjugacy classes — view the full character table.
| maximal subgroup of | #64862, C4×C8×D5, #64974, … |
| maximal normal subgroup of | #64862, C4×C8×D5, #64974, … |
| maximal quotient of | #64937, #64938, D5×#95, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 92);