Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#381 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C23 |
| commutator | C22 | C22×C4×C5 |
| Frattini, cosocle | C23 | C23×C5 |
| prime radical | C24×C5 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,19,20,21);(10,13)(14,16);(6,8)(9,11,12,15)(10,14,13,16)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 882);