Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C33 ⋊ #365 |
114 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=c,b^2=cef,c^2=f,d^2=ef,e^2=f,b^a=bcde,d^a=de,e^a=ef,g^a=g^2,h^a=i,i^a=h^2,c^b=cf,d^b=def,e^b=ef,h^b=hi,i^b=hi^2,h^c=h^2,i^c=i^254 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 21210);