Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| isoclinism | isoclinic class of #1437⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#1437⋊C3 |
| unambiguous semidirect product | #1437⋊(C3×C5) |
| unambiguous semidirect product | (C5×#1437)⋊C3 |
| Fitting, Hall | C5×#1437 ⋊ C3 |
| derived | C22×C42 ⋊ C2×C3×C5 |
| Sylow | #1437 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | #1437⋊C3 |
| commutator, cosocle, hypocenter | C22×C42 | C2×C3×C5 |
| Frattini | C24 | C2×C5×A4 |
| Fitting | C5×#1437 | C3 |
| socle | C24×C5 | C2×A4 |
| LCS | G ▷ C22×C42 | |
| UCS | C1 ◁ C5 | |
336 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 26 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)(6,7)(8,12,13,10,9,11)(14,15,23)(16,21,25)(17,18,19)(20,24,22);(7,11)(10,12)(14,16,17,22)(15,18)(19,24,23,25)(20,21)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 236365);