Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic3⋊Dic27 |
| Sylow | C3×C27 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×D27 |
| commutator | C2×C3×C27 | C2×C4 |
| Frattini | C22×C9 | D32 |
| Fitting, cosocle | C22×C3×C27 | C22 |
| socle, prime radical | C22×C32 | C2×D9 |
| hypocenter | C3×C27 | C4⋊C4 |
| LCS | G ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C22 | |
128 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^4,3^4|a^2=d,b^2=c,e^3=g,g^3=h,b^a=bd,f^a=f^2,e^b=e^2g^2h^2,g^b=g^2h^2,h^b=h^2174 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 194);