Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92 · Schur multiplier (H2) = C32×C9 |
| direct product | C9×#6873 |
| subgroup | G/H | |
|---|---|---|
| center | C92 | H(3,𝔽3) |
| commutator | C32 | C3×C92 |
| Frattini, cosocle | C32×C9 | C33 |
| socle | C32 | C3×C32⋊C9 |
| prime radical | C3×H(3,𝔽3) | C3×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C92 ◁ C3×C92 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 90 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 3 generators on 90 points:
(1,2,3,4,5,6,7,8,9);(10,11,15,13,18,30,25,36,52,27,17,33,48,35,57,71,60,78,14,19,29,26,37,51,47,61,74)(12,20,38,23,40,63,45,65,82,22,16,28,44,32,50,69,55,73,24,34,56,46,59,77,70,80,88)(21,39,49,42,64,72,67,83,86,43,31,53,68,54,75,85,76,87,41,58,62,66,79,81,84,89,90);(10,12,21)(11,16,31)(13,23,42)(14,24,41)(15,28,49)(17,34,58)(18,32,54)(19,20,39)(22,43,27)(25,45,67)(26,46,66)(29,38,62)(30,50,72)(33,56,53)(35,59,79)(36,55,76)(37,40,64)(44,68,48)(47,70,84)(51,63,81)(52,73,86)(57,77,75)(60,80,89)(61,65,83)(69,85,71)(74,82,90)(78,88,87)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4118);