Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#339 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | D4 |
| commutator | C4 | C42×C5 |
| Frattini, cosocle | C22×C4 | C22×C5 |
| socle | C22×C5 | C22⋊C4 |
| prime radical | C23×C5 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ G | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,4)(5,6)(7,9)(8,11)(10,12)(13,14,15,16,17);(6,9,11,10)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 144);