Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ C2×#99 |
787 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 787 conjugacy classes of subgroups — too many to draw. Only its 101 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|b^2=e,g^a=g^2,i^a=i^2,c^b=cd,d^b=df,h^b=h^2,i^b=i^2,e^c=ef,h^g=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72220471, #72252896, #72252959, … |
| maximal normal subgroup of | #72252896, #72252959, #99×((C3⋊D3)⋊C3)⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 6593);