Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #64210 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C5⋊S4 |
| commutator, cosocle | C2×C3×C5×A4 | C22 |
| Fitting, socle | C23×C3×C5 | C2×D3 |
| hypocenter | C3×C5×A4 | D4 |
| LCS | G ▷ C2×C3×C5×A4 ▷ C3×C5×A4 | |
| UCS | C1 ◁ C2 | |
268 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 34 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:
2^3,3^2,5,2^2|a^2=c,b^a=bc,d^a=d^2,f^a=f^4,g^a=gh,e^b=e^2,g^d=h,h^d=gh72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4862);