Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C3×C5×#1406 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C5 | #353 |
| commutator | C42 | C2×C3×C4×C5 |
| Frattini, cosocle | #124 | C22×C3×C5 |
| prime radical | C3×C5×#496 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C5 ◁ C22×C3×C5 ◁ C2×C3×C4×C5 ◁ C3×C5×#124 ◁ G | |
216 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10,11,16)(12,15,18,23)(13,17,19,24)(14,20,22,21);(10,14)(11,18)(12,21)(15,17)(19,20)(22,23)210 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 138);