Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#88391 |
| derived | C3×Q8⋊C27 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊((C22⋊C27)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C27 | C2 |
| Frattini | C2×C9 | C3⋊S4 |
| Fitting | C3×Q8×C9 | D3 |
| socle | C2×C32 | (C22⋊C9)⋊C2 |
| LCS | G ▷ C3×Q8⋊C27 | |
| UCS | C1 ◁ C2 | |
118 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2,3^4,2^3|b^3=d,d^3=e,f^2=h,g^2=h,b^a=b^2d^2e^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, 704);