Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C33×Q8 ⋊ D4 |
| Sylow | C33 ⋊ #449 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11724331 |
| commutator, cosocle | C3×C4×C3⋊D3 | C23 |
| Frattini | C4 | C2×D32⋊D3 |
| Fitting | C33×Q8 | D4 |
| socle | C2×C33 | #120 |
| hypercenter | Q8 | D32⋊D3 |
| hypocenter | C33 | #449 |
| LCS | G ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
616 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 616 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|c^2=f,e^2=f,b^a=bd,c^a=ce,e^a=ef,g^a=g^2,h^a=h^2,i^a=h^2i,c^b=cf,h^b=h^2i^2,e^c=ef,h^d=h^2,i^d=i^281 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 33312);