Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C5 · Schur multiplier (H2) = C26 |
| direct product | C3×C5×#3025 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C5 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C3×C5 |
| socle | C2×C3×C5 | C2×#120 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C5 ◁ C22×C3×C4×C5 ◁ G | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3)(4,5,6,7,8)(14,20)(21,24);(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)480 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 1757);