Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #4174⋊Q8 |
| Sylow | AΣL(1,𝔽27) ⋊ C2×Q8 |
316 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 42 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:
2^4,3^4|b^2=d,c^2=d,e^a=e^2,g^a=g^2h,c^b=cd,f^b=f^2,g^b=g^2,h^b=h^2,f^e=fg,g^e=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 828);