Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#2032⋊C3 |
| unambiguous semidirect product | #2032⋊(C3×C5) |
| unambiguous semidirect product | (C5×#2032)⋊C3 |
| Fitting, Hall | C5×#2032 ⋊ C3 |
| derived | #543 ⋊ C2×C3×C5 |
| Sylow | #2032 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #393⋊C3 |
| commutator, cosocle, hypocenter | #543 | C2×C3×C5 |
| Frattini | C24 | C2×C5×A4 |
| Fitting | C5×#2032 | C3 |
| hypercenter | C22×C5 | #120⋊C3 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
166 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7)(8,16,17,11,9,21)(10,13)(12,14,20,18,19,15);(6,14)(7,21,13,19)(8,9)(10,17)(11,12,20,16)(15,18)105 conjugacy classes — view the full character table.
| maximal subgroup of | C5×#72209831, D5×#2032⋊C3 |
| maximal normal subgroup of | D5×#2032⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236392);