Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #576 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#157⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C24 |
| Fitting | C33×Q8 | D4 |
| socle | C2×C33 | C22×D4 |
| hypercenter | Q8 | #157⋊D3 |
| hypocenter | C33 | #576 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
948 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|c^2=f,d^2=f,e^2=f,b^a=be,d^a=df,e^a=ef,g^a=g^2,h^a=h^2,i^a=h^2i,d^b=df,e^b=ef,g^b=g^2,h^b=h^2i^2,d^c=df,h^e=h^2,i^e=i^272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47220);