Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#340 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | D4 |
| commutator | C4 | C42×C5 |
| Frattini, cosocle | C22×C4 | C22×C5 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C4×C5 ◁ G | |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4)(10,14)(11,12)(15,16)(17,18,19,20,21);(5,6,7,8)(9,10,12,15,13,16,11,14)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 145);