Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C93 · Schur multiplier (H2) = C3×C92 |
| direct product | C9×#6864 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C92 | C32 |
| commutator | C3 | C93 |
| Frattini, cosocle | C32×C9 | C33 |
| socle, prime radical | C33 | C32×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C92 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2187 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9);(19,20,23,21,25,31,27,34,39,22,26,30,28,35,38,36,42,44,29,24,32,37,33,40,43,41,45);(10,11,13,12,14,16,15,17,18)(20,24,26)(23,30,32)(25,33,35)(31,38,40)(34,41,42)(39,44,45)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4071);