Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #443 |
401 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|c^2=d,d^2=f,e^2=f,b^a=be,c^a=cf,e^a=ef,g^a=g^2,i^a=i^2,c^b=ce,e^b=ef,h^b=h^2,i^b=i^2,e^c=ef,h^g=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 5231);