Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #99 |
| commutator | C23 | C2×C4×C5 |
| Frattini, cosocle | C2×D4 | C22×C5 |
| prime radical | C2×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ G | |
68 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)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)(17,18,19,20,21);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 157);