Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | H(3,𝔽3)×#126 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #560 |
942 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 942 conjugacy classes of subgroups — too many to draw. Only its 202 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:
2^4,3,2^2,3^2|c^2=g,d^2=f,c^a=cg,d^a=dfg,h^a=h^2,i^a=i^2,c^b=cf,d^b=dg,d^c=dg,h^e=hi113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19304);