Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C9⋊GL(2,𝔽3)) |
| unambiguous semidirect product | C9⋊(C3⋊GL(2,𝔽3)) |
| Fitting | C3×Q8×C9 ⋊ D3 |
| derived | C3×C9×SL(2,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C9⋊S4) |
| commutator, cosocle, hypocenter | C3×C9×SL(2,𝔽3) | C2 |
| Frattini | C2×C3 | C3⋊(C3⋊S4) |
| Fitting | C3×Q8×C9 | D3 |
| socle | C2×C32 | C3⋊S4 |
| LCS | G ▷ C3×C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
370 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 71 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:
2,3^4,2^3|d^3=e,f^2=h,g^2=h,b^a=b^2,c^a=c^2,d^a=d^2e^2,e^a=e^2,f^a=fg,g^a=gh,f^b=g,g^b=fgh,g^f=gh99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2133);