Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C36 |
| direct product | C32×#6873 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C3×C9 | C34 |
| socle | C33 | C32⋊C9 |
| prime radical | C32×H(3,𝔽3) | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2,3);(4,5,6);(7,8,12,10,15,27,22,33,49,24,14,30,45,32,54,68,57,75,11,16,26,23,34,48,44,58,71)(9,17,35,20,37,60,42,62,79,19,13,25,41,29,47,66,52,70,21,31,53,43,56,74,67,77,85)(18,36,46,39,61,69,64,80,83,40,28,50,65,51,72,82,73,84,38,55,59,63,76,78,81,86,87);(7,9,18)(8,13,28)(10,20,39)(11,21,38)(12,25,46)(14,31,55)(15,29,51)(16,17,36)(19,40,24)(22,42,64)(23,43,63)(26,35,59)(27,47,69)(30,53,50)(32,56,76)(33,52,73)(34,37,61)(41,65,45)(44,67,81)(48,60,78)(49,70,83)(54,74,72)(57,77,86)(58,62,80)(66,82,68)(71,79,87)(75,85,84)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 7992);