Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#356 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #99 |
| commutator | C2×C4 | C2×C4×C5 |
| Frattini, cosocle | C2×Q8 | C22×C5 |
| prime radical | C22×C5 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×Q8 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,5,9,6,10,3,11)(4,8,12,15,14,16,13,7)(17,18,19,20,21);(1,3,6,5)(2,7,10,15)(4,13,14,12)(8,9,16,11)65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 161);