Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#410 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22⋊C4 |
| commutator | C22 | C22×C4×C5 |
| Frattini, cosocle | C23 | C23×C5 |
| socle | C2×C5 | C2×C22⋊C4 |
| prime radical | C2×C5×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,19,20,21);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 911);