Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C33 ⋊ #469 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×#157⋊D3 |
| commutator, cosocle | C2×C3×C3⋊Dic3 | C23 |
| Fitting | C23×C33 | D4 |
| socle | C22×C33 | C2×D4 |
| hypercenter | C23 | #157⋊D3 |
| hypocenter | C33 | #469 |
| LCS | G ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
556 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 55 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|d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=g^2,h^a=h^2,i^a=h^2i,c^b=cef,d^b=df,g^b=g^2,h^b=h^2i^2,h^d=h^2,i^d=i^272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 33658);