Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#2679⋊C3 |
| unambiguous semidirect product | #2679⋊(C3×C5) |
| unambiguous semidirect product | (C5×#2679)⋊C3 |
| Fitting, Hall | C5×#2679 ⋊ C3 |
| derived, Sylow | #2679 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | #5464 |
| commutator, cosocle, hypocenter | #2679 | C3×C5 |
| Frattini | C23 | C5×C22⋊A4 |
| Fitting | C5×#2679 | C3 |
| socle | C23×C5 | C22⋊A4 |
| LCS | G ▷ #2679 | |
| UCS | C1 ◁ C2×C5 | |
430 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4,5)(6,7,10)(8,11,16)(9,13,15)(12,17,14);(8,14)(11,12);(6,14,9,8)(10,16,15,17)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236709);