Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×D4×H(3,𝔽3) ⋊ C4 |
| Sylow | H(3,𝔽3) ⋊ #421 |
559 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 559 conjugacy classes of subgroups — too many to draw. Only its 57 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|a^2=d,e^2=f,b^a=bf,c^a=ce,g^a=gh,h^a=gh^2i^2,d^c=df,e^c=ef,g^d=g^2,h^d=h^2i,h^g=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13463);