Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #11722281⋊C2 |
| Sylow | C22×#41 ⋊ 3−1+2 |
241 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 241 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.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,b^3=d,c^3=d,g^2=i,h^2=i,c^b=cd,g^b=gh,h^b=g,e^c=ef,f^c=e,h^g=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30345);