Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C27 · Schur multiplier (H2) = C36 |
| direct product | C32×C9⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center | C33×C9 | C32 |
| commutator | C3 | C33×C27 |
| Frattini, cosocle | C3×C9 | C34 |
| socle, prime radical | C34 | C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
1196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 2187 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2,3);(4,5,6);(8,10,13)(11,15,14)(16,17,20,18,21,26,23,28,33,19,22,27,24,29,34,31,36,39,25,30,35,32,37,40,38,41,42);(7,8,11,9,10,14,12,13,15)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 8003);