Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #362 |
219 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=c,c^2=ef,d^2=ef,e^2=f,b^a=bd,c^a=ce,d^a=de,e^a=ef,h^a=h^2,i^a=i^2,d^b=df,g^b=g^2,i^b=i^2,h^g=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1139);