Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×C27 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10619544 |
| commutator, Fitting, cosocle | C3×C4×C27 | C22 |
| Frattini | C4×C9 | D32 |
| socle | C2×C32 | #611 |
| hypercenter | C4 | D3×D27 |
| hypocenter | C3×C27 | D8 |
| LCS | G ▷ C3×C4×C27 ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C2 ◁ C4 | |
100 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 100 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^4,3^4|c^2=d,f^3=g,g^3=h,b^a=bc,c^a=cd,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=cd,e^b=e^2129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 119);