Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | H(3,𝔽3)×#135 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #578 |
1364 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 1364 conjugacy classes of subgroups — too many to draw. Only its 199 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|c^2=f,d^2=f,e^2=f,b^a=bf,g^a=g^2,h^a=h^2,c^b=cf,d^b=df,d^c=de,e^c=ef,e^d=ef,h^g=hi122 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20439);