Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C43×C5 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×C4×C5×#97 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C42×C5 | C22 |
| commutator | C2 | C3×C43×C5 |
| Frattini, cosocle | C22×C4 | C23×C3×C5 |
| socle, prime radical | C23×C3×C5 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C42×C5 ◁ G | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 1920 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3)(4,5,6,7)(20,21,22,23,24);(12,13,14,17,15,16,18,19);(8,9,10,11)(13,16)(17,19)1200 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 457);