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) = C2×C4 |
| direct product | C5×H(3,ℤ/4) |
| 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 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
94 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)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,19,20,21);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 142);