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) = C22 |
| direct product | C5×#331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C5 |
| Frattini, cosocle | C42 | C22×C5 |
| prime radical | C5×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
84 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,6,4,5,7,8)(9,10)(11,13)(12,15)(14,16)(17,18,19,20,21);(2,5)(3,7)(10,13)(11,16)(14,15)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 136);