Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#368 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | D8 |
| commutator | C8 | C2×C4×C5 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| socle | C2×C5 | #107 |
| prime radical | C22×C5 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×C8 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,5,10,7,11,6,21)(3,12,15,27,17,28,16,31)(4,18,19,25,20,32,13,9)(8,22,23,30,24,29,26,14)(33,34,35,36,37);(1,3,13,29,6,16,20,30,7,17,19,22,5,15,4,14)(2,8,18,27,10,23,25,28,11,24,32,31,21,26,9,12)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 173);