Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C92 · Schur multiplier (H2) = C34×C9 |
| direct product | C9×#6903 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C27 | C32 |
| commutator | C3 | C32×C92 |
| Frattini, cosocle | C3×C9 | C34 |
| socle | C32 | C33×C9 |
| prime radical | C3×H(3,𝔽3) | C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C27 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 90 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 90 points:
(1,2,3,4,5,6,7,8,9);(10,11,15,13,17,27,23,33,48,14,18,28,24,34,49,42,57,71,25,35,50,43,58,72,64,78,86)(12,16,26,21,31,46,40,55,69,22,32,47,41,56,70,63,77,85,20,30,45,39,54,68,62,76,84)(19,29,44,37,52,66,60,74,82,36,51,65,59,73,81,79,87,89,38,53,67,61,75,83,80,88,90);(10,12,19)(11,16,29)(13,21,37)(14,22,36)(15,26,44)(17,31,52)(18,32,51)(20,38,25)(23,40,60)(24,41,59)(27,46,66)(28,47,65)(30,53,35)(33,55,74)(34,56,73)(39,61,43)(42,63,79)(45,67,50)(48,69,82)(49,70,81)(54,75,58)(57,77,87)(62,80,64)(68,83,72)(71,85,89)(76,88,78)(84,90,86);(12,20,22)(16,30,32)(19,36,38)(21,39,41)(26,45,47)(29,51,53)(31,54,56)(37,59,61)(40,62,63)(44,65,67)(46,68,70)(52,73,75)(55,76,77)(60,79,80)(66,81,83)(69,84,85)(74,87,88)(82,89,90)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 7962);