Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#144366 (order 10240) · Inn(G) = C2×D5 · Out(G) = #233538 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C22⋊Dic5 |
| Sylow | C5 ⋊ C2×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×D5 |
| commutator | C2×C5 | C22×C4 |
| Frattini | C22 | C22×D5 |
| Fitting, prime radical | C24×C5 | C2 |
| socle | C23×C5 | C22 |
| cosocle | C22×C5 | C23 |
| hypercenter | C24 | D5 |
| hypocenter | C5 | C2×C22⋊C4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ C24 | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 15 points:
(3,4)(5,6)(11,12,14,15,13);(4,6)(7,8,9,10)(12,13)(14,15);(1,2)52 conjugacy classes — view the full character table.
| maximal subgroup of | #64746, #64747, #64933, … |
| maximal normal subgroup of | #64746, #64747, #64933, … |
| maximal quotient of | C2×C22⋊#168, #65496, C2×#3963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 173);