Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #85647 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#85647 |
| Fitting, Hall | C22×C5×#127 ⋊ C3 |
| derived | C22×C42 ⋊ C2×C3×C5 |
| Sylow | C22×#127 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #85647 |
| commutator, cosocle, hypocenter | C22×C42 | C2×C3×C5 |
| Frattini | C22 | C2×C5×C22⋊A4 |
| Fitting | C22×C5×#127 | C3 |
| socle | C24×C5 | C2×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C5 | |
1080 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1080 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) = 21 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(7,8,9)(10,11,14)(12,16,19,17,18,21)(13,15,20);(6,9)(7,8)(11,15)(14,20)(16,18)(19,21);(10,17,13,12)(11,16,15,18)(14,20)(19,21)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237083);