Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×C5×#1297 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C5 | C22⋊C4 |
| commutator | C2×C4 | C3×C42×C5 |
| Frattini, cosocle | C23×C4 | C22×C3×C5 |
| socle | C22×C3×C5 | #95 |
| prime radical | C24×C3×C5 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C5 ◁ C23×C3×C4×C5 ◁ G | |
492 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 480 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3)(4,5,6,7,8)(9,10,12,15,13,16,11,14)(17,18)(19,23)(20,22)(21,24);(10,14,16,15)(17,19)(18,21,22,23)(20,24)570 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 29);