Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| isoclinism | isoclinic class of C3⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊(A4⋊#168) |
| unambiguous semidirect product | A4⋊(C3⋊#168) |
| derived | C3×C5×A4 ⋊ C8 |
| Sylow | C5 ⋊ C3⋊(A4⋊C8) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | C3⋊(C5⋊S4) |
| commutator, hypocenter | C3×C5×A4 | C8 |
| Fitting | C22×C3×C4×C5 | D3 |
| socle | C23×C3×C5 | Dic3 |
| prime radical | C2×C3×C5×A4 | C4 |
| cosocle | C3×C4×C5×A4 | C2 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C4 | |
136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^3,3^2,5,2^2|a^2=b,b^2=c,d^a=d^2,e^a=e^2,f^a=f^4,g^a=gh,g^d=h,h^d=gh132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 2534);