Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C24×H(3,𝔽3) ⋊ C22 |
| Sylow | H(3,𝔽3) ⋊ C2×#120 |
2447 conjugacy classes of subgroups; only the 215 normal subgroups are shown.
This group has 2447 conjugacy classes of subgroups — too many to draw. Only its 215 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|g^a=g^2,i^a=i^2,c^b=cf,d^b=de,h^b=h^2,i^b=i^2,h^g=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72220439, #120×((C3⋊D3)⋊C3)⋊C2, #72252959 |
| maximal normal subgroup of | #120×((C3⋊D3)⋊C3)⋊C2, #72252959 |
| maximal quotient of | #72252896 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18156);