Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C3⋊(C5⋊(#127⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | C3⋊(C5⋊(#127⋊C3)) |
| unambiguous semidirect product | C5⋊(C3⋊(#127⋊C3)) |
| unambiguous semidirect product | #91763⋊C3 |
| Fitting, derived | C3×C42×C5 ⋊ C2×C3 |
| Sylow | C5 ⋊ C3⋊(#127⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C42×C5 | C2×C3 |
| Frattini | C22 | A4×D15 |
| socle | C22×C3×C5 | C2×A4 |
| LCS | G ▷ C3×C42×C5 | |
| UCS | C1 | |
118 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 118 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2,3,2^2,3,5,2^2|c^2=h,d^2=gh,c^a=ch,d^a=dgh,e^a=e^2,f^a=f^4,c^b=d,d^b=cdg,g^b=h,h^b=gh66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2552);