Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #64204 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×D15 |
| commutator, prime radical, cosocle | C2×C3×C5×A4 | C22 |
| Fitting, socle | C23×C3×C5 | C2×D3 |
| hypocenter | C3×C5×A4 | Q8 |
| LCS | G ▷ C2×C3×C5×A4 ▷ C3×C5×A4 | |
| UCS | C1 ◁ C2 | |
180 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^3,3^2,5,2^2|a^2=c,b^2=c,b^a=bc,e^a=e^2,f^a=f^4,d^b=d^2,g^b=gh,g^d=h,h^d=gh84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4871);