Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C3 |
No aliases for this group.
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3×C3⋊D3 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C24×C33 | C22 |
| hypercenter | C24 | D3×C3⋊D3 |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C24 | |
6964 conjugacy classes of subgroups; only the 336 normal subgroups are shown.
This group has 6964 conjugacy classes of subgroups — too many to draw. Only its 336 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^6,3^3|c^a=cf,d^a=de,g^a=g^2,h^a=h^2,c^b=cf,d^b=de,i^b=i^2162 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊D3×#120, D3×#10595527, #72220612, … |
| maximal normal subgroup of | D3×C3⋊D3×#120, D3×#10595527, #72220666, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44171);