Finite Group DatabasebetaFinite Group Databasebeta| order | 1440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ #64201 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S4×D15 |
| commutator, cosocle, hypocenter | C3×C5×SL(2,𝔽3) | C22 |
| Fitting | C3×C5×Q8 | C2×D3 |
| socle | C2×C3×C5 | C2×S4 |
| LCS | G ▷ C3×C5×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
166 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^2,3^2,5,2^3|f^2=h,g^2=h,b^a=bh,d^a=d^2,e^a=e^4,c^b=c^2,f^b=fg,g^b=gh,f^c=g,g^c=fgh,g^f=gh69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1440, 4868);