Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | C22×C4×#732 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | H(3,𝔽3) |
| commutator | C32 | C22×C32×C4 |
| Frattini, cosocle | C2×C32 | C23×C32 |
| socle | C23×C3 | C2×H(3,𝔽3) |
| prime radical | C23×#732 | C2 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C22×C32×C4 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(9,10,13)(11,17,21)(12,16,22)(14,23,19)(15,26,20)(18,28,31)(24,34,35)(25,33,30)(27,32,29);(3,4)(9,11,18,20,30,35,12,19,29)(10,14,24,15,17,27,16,25,28)(13,21,31,26,33,34,22,23,32);(5,6,7,8)272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1208);