Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5×C82 · Schur multiplier (H2) = C4 |
| direct product | C3×C5×#1311 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5×C8 | C22 |
| commutator | C2 | C3×C5×C82 |
| Frattini, cosocle | C4×C8 | C22×C3×C5 |
| socle, prime radical | C22×C3×C5 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5×C8 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 1920 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3)(4,5,6,7,8)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32);(9,10,11,13,12,14,15,16)(18,22)(23,28)(24,29)(30,32)1200 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 43);