Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C16 · Schur multiplier (H2) = C2 |
| direct product | C5×C4⋊C16 |
| unambiguous semidirect product | C4⋊(C5×C16) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C22 |
| commutator | C2 | C2×C5×C16 |
| Frattini, cosocle | C2×C8 | C22×C5 |
| socle, prime radical | C22×C5 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,4)(5,6,7,10,8,11,13,16,9,12,14,17,15,18,19,20)(21,22,23,24,25);(1,2,3,4)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 168);