Finite Group DatabasebetaFinite Group Databasebeta| order | 1920 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C1 |
| direct product | SL(2,𝔽3)×#674 |
| unambiguous semidirect product | (Q8×#674)⋊C3 |
| derived | C5×Q8 ⋊ C3×C16 |
| Hall | Q8×#674 ⋊ C3 |
| Sylow | C5 ⋊ C16×SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | A4×AGL(1,𝔽5) |
| commutator, hypocenter | C5×Q8 | C3×C16 |
| Fitting | C4×C5×Q8 | C3×C4 |
| socle | C22×C5 | C8×A4 |
| prime radical | C2×C5×SL(2,𝔽3) | C8 |
| cosocle | Q8×#168 | C2×C3 |
| LCS | G ▷ C5×Q8 | |
| UCS | C1 ◁ C2×C4 | |
108 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1920, 237298);