Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#2273⋊C3 |
| unambiguous semidirect product | #2273⋊(C3×C5) |
| unambiguous semidirect product | (C5×#2273)⋊C3 |
| Fitting, Hall | C5×#2273 ⋊ C3 |
| derived, Sylow | #2273 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | C22⋊(C42⋊C3) |
| commutator, cosocle, hypocenter | #2273 | C3×C5 |
| Frattini | C23 | C5×C22⋊A4 |
| Fitting | C5×#2273 | C3 |
| socle | C23×C5 | C22⋊A4 |
| LCS | G ▷ #2273 | |
| UCS | C1 ◁ C2×C5 | |
374 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(7,8,10)(11,12,13)(14,15,18)(16,19,24)(17,21,23)(20,25,22);(7,11)(8,12);(6,10)(7,8)(9,13)(11,12)(14,22,17,16)(18,24,23,25)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236707);