Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×C9 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11720239 |
| commutator, Fitting, cosocle | C3×C9×C16 | C22 |
| Frattini | C3×C16 | D32 |
| socle | C2×C32 | #826 |
| hypercenter | C16 | D3×D9 |
| hypocenter | C3×C9 | D32 |
| LCS | G ▷ C3×C9×C16 ▷ C3×C8×C9 ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
130 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|b^2=cd,c^2=de,d^2=ef,e^2=f,g^3=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,h^a=h^2,g^b=g^2i^2,i^b=i^2159 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 965);