Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#338 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C42 |
| commutator | C4 | C42×C5 |
| Frattini, cosocle | C22×C4 | C22×C5 |
| socle | C2×C5 | #95 |
| prime radical | C23×C5 | Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,8,6,9,5,14)(3,10,12,7,13,15,11,16)(17,18,19,20,21);(1,3,5,12,6,13,4,11)(2,7,8,10,9,16,14,15)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 143);