Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #102 |
| commutator | C8 | C2×C4×C5 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| prime radical | C5×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×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:
(2,6,7,15)(3,9)(4,12)(8,10,13,16)(11,14)(17,18,19,20,21);(1,2)(3,10)(4,13)(5,7)(6,14)(8,12)(9,15)(11,16)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 165);