Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived | C9×SL(2,𝔽3) ⋊ C2×C4 |
321 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^3,3^3,2^3|a^2=c,b^2=i,e^3=f,g^2=i,h^2=i,b^a=bi,d^a=d^2,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,g^d=h,h^d=ghi,h^g=hi138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13045);