Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | B(4,𝔽2)⋊C27 |
| derived | H(5,𝔽2) ⋊ C2×C27 |
| Sylow | B(4,𝔽2) ⋊ C27 |
144 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|b^3=c,c^3=d,e^a=eg,f^a=fgh,e^b=ef,f^b=e,g^b=hi,h^b=ghi,h^e=hi,g^f=gi,h^f=hi144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 210);