Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#398 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C23 |
| commutator, Frattini, cosocle | C23 | C23×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4)(9,10)(11,16)(12,14)(13,15)(17,18,19,20,21);(5,6)(7,8)(9,11,13,12)(10,14,15,16);(2,4)(6,8)(9,12,13,11)(10,14,15,16)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 899);