Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3×C27 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D3×D27 |
| commutator, cosocle | C2×C3×C27 | C23 |
| Frattini | C2×C9 | C2×D32 |
| Fitting | C3×C4×C27 | C22 |
| socle | C2×C32 | C22×D9 |
| hypercenter | C4 | D3×D27 |
| hypocenter | C3×C27 | #41 |
| LCS | G ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C2 ◁ C4 | |
178 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 50 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|b^2=d,c^2=d,f^3=g,g^3=h,c^a=cd,e^a=e^2,f^b=f^2g^2h^2,g^b=g^2h^2,h^b=h^2135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 820);