Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2812⋊C3 |
| unambiguous semidirect product | #2812⋊(C3×C5) |
| unambiguous semidirect product | (C5×#2812)⋊C3 |
| Fitting, Hall | C5×#2812 ⋊ C3 |
| Sylow | #2812 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | #5450 |
| commutator, cosocle | C25 | C22×C3×C5 |
| Frattini | C23 | C22×C5×A4 |
| Fitting | C5×#2812 | C3 |
| socle | C23×C5 | C22×A4 |
| hypocenter | C24 | C3×C5×D4 |
| LCS | G ▷ C25 ▷ C24 | |
| UCS | C1 ◁ C2×C5 | |
324 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 40 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 — 2 generators on 25 points:
(1,2,3,4,5)(7,9)(10,12)(11,16)(13,19,14,22,15,20)(17,23,18,24,21,25);(6,7)(8,9)(10,18,15,11)(12,24,19,23)(13,17,14,21)(16,22,25,20)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236626);