Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | H(3,𝔽3)×#120 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ B(4,𝔽2) |
1089 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 99 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=ce,d^a=df,g^a=g^2,h^a=h^2,c^b=cd,e^b=ef,h^g=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72252959, #272×B(4,𝔽2) |
| maximal normal subgroup of | #72252959, #272×B(4,𝔽2) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10512);