Finite Group DatabasebetaFinite Group Databasebeta| order | 720 |
| isoclinism | isoclinic class of #6758 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×C4⋊ΓL(1,𝔽16) |
| unambiguous semidirect product | C4⋊(C3×ΓL(1,𝔽16)) |
| Fitting | C32×C4×C5 ⋊ C4 |
| Hall | C32×C5 ⋊ C4⋊C4 |
| Sylow | C32 ⋊ C4⋊AGL(1,𝔽5) |
| Sylow | C5 ⋊ C3×C4⋊Dic3 |
120 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#91152, C3×#91154, C3×#91159, … |
| maximal normal subgroup of | C3×#91152, C3×#91154, C3×#91159, … |
| maximal quotient of | C3×#91222, C3×#91223, C3×#91224, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(720, 557);