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) = C22 |
| direct product | C5×#436 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C23 |
| commutator | C22 | C22×C4×C5 |
| Frattini, cosocle | C2×C4 | C23×C5 |
| socle | C22×C5 | C22×C4 |
| prime radical | C2×C5×D4 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(3,4)(5,6,9,7,8,10,12,11)(13,14,15,16,17);(2,4)(6,10)(7,11);(5,7,9,10,8,11,12,6)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 937);