Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C27 · Schur multiplier (H2) = C32×C9 |
| direct product | C27×C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C27 | C32 |
| commutator | C3 | C3×C9×C27 |
| Frattini, cosocle | C32×C9 | C33 |
| socle, prime radical | C33 | C32×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C27 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C3 ▷ C1 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2187 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(29,31,34)(32,36,35)(37,38,40,39,41,43,42,44,45);(28,29,32,30,31,35,33,34,36)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4127);