Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | ((C3⋊S4)⋊C3)⋊C8 |
| derived | C22⋊H(3,𝔽3) ⋊ C2×C8 |
320 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=c,c^2=d,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=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12346);