Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C16 · Schur multiplier (H2) = C3 |
| direct product | C16×C9⋊C9 |
| unambiguous semidirect product | (C9×C16)⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C16 | C32 |
| commutator | C3 | C3×C9×C16 |
| Frattini, cosocle | C32×C8 | C2×C32 |
| socle, prime radical | C2×C32 | C32×C8 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C16 ◁ G | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 1296 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(18,20,23)(21,25,24)(26,27,29,28,30,32,31,33,34);(17,18,21,19,20,24,22,23,25)528 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 50);