Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C3⋊D27)⋊Q8 |
| Sylow | C3×C27 ⋊ C2×Q8 |
176 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 52 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:
2^4,3^4|a^2=d,b^2=d,c^2=d,e^3=g,g^3=h,c^a=cd,e^a=e^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=cd,f^b=f^2132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 817);