Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#330 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C5 |
| Frattini, cosocle | C42 | C22×C5 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,5,6,7)(9,10,12,14,13,15,11,16)(17,18,19,20,21);(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,12,14)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 135);