Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×C4⋊(#120⋊C3) |
| unambiguous semidirect product | C4⋊(C5×#120⋊C3) |
| unambiguous semidirect product | #2847⋊(C3×C5) |
| unambiguous semidirect product | (C5×#2847)⋊C3 |
| Fitting, Hall | C5×#2847 ⋊ C3 |
| Sylow | #2847 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×#120⋊C3 |
| commutator, cosocle | C25 | C22×C3×C5 |
| Frattini | C23 | C22×C5×A4 |
| Fitting | C5×#2847 | C3 |
| socle | C23×C5 | C22×A4 |
| hypercenter | C4×C5 | #120⋊C3 |
| hypocenter | C24 | C3×C5×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
508 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 120 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,3,4,5)(6,7,8,9)(12,13,14)(15,16,17);(7,9)(10,17,12,11)(13,16,14,15)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236637);