Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C2 |
| direct product | C5×#349 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22⋊C4 |
| commutator | C22 | C2×C5×C8 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| socle | C2×C5 | C22⋊C8 |
| prime radical | C23×C5 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×C8 ◁ G | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,7,4,8,10,14,5,6,9,12,11,13,15,16)(17,18,19,20,21);(2,6)(3,9)(8,13)(10,15)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 154);