Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | H(3,𝔽3)×Q32 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #508 |
462 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 462 conjugacy classes of subgroups — too many to draw. Only its 71 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|b^2=f,c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^2,h^a=h^2,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef,h^g=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 10286);