Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#85567 |
| derived | #997 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C5 | #1020 |
| commutator, hypocenter | #997 | C4×C5 |
| Frattini | C4 | C5×#1020 |
| Fitting | C5×#585 | D3 |
| socle | C2×C5 | #5936 |
| prime radical | C5×#5950 | C2 |
| cosocle | #5950 | C2×C5 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C4×C5 | |
330 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,13,33,26)(7,16,36,23)(8,25,24,19)(9,29,17,27)(10,15,30,37)(11,22,35,32)(12,21,31,28)(14,18,34,20);(9,20,10)(13,29,14)(16,33,17)(19,22,31)(21,24,23)(25,36,26)(28,30,35)(32,34,37)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237003);