Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
294 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=c,c^2=d,b^a=bd,f^a=f^2,g^a=g^2,e^b=e^2,g^b=g^2,h^b=hi,f^e=fg,h^e=i,i^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12349);