Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C16 · Schur multiplier (H2) = C32 |
| direct product | C16×#732 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C16 | H(3,𝔽3) |
| commutator | C32 | C32×C16 |
| Frattini, cosocle | C32×C8 | C2×C32 |
| socle | C2×C3 | C8×H(3,𝔽3) |
| prime radical | C2×#732 | C8 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C16 ◁ C32×C16 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,21)(19,25,29)(20,24,30)(22,31,27)(23,34,28)(26,36,39)(32,42,43)(33,41,38)(35,40,37);(17,19,26,28,38,43,20,27,37)(18,22,32,23,25,35,24,33,36)(21,29,39,34,41,42,30,31,40)272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 55);