Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2823⋊C3 |
| unambiguous semidirect product | #2823⋊(C3×C5) |
| unambiguous semidirect product | (C5×#2823)⋊C3 |
| Fitting, Hall | C5×#2823 ⋊ C3 |
| Sylow | #2823 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | #560⋊C3 |
| commutator, cosocle | C2×C42 | C22×C3×C5 |
| Frattini | C23 | C22×C5×A4 |
| Fitting | C5×#2823 | C3 |
| socle | C23×C5 | C22×A4 |
| hypocenter | C42 | C3×C5×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2×C5 | |
232 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 232 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) = 240 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(7,9)(10,11,12)(13,23,22,21,24,19)(14,16,20,17,18,25);(6,7)(8,9)(10,20,12,25)(11,24,15,23)(13,22,18,17)(14,16,19,21)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236628);