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 | C9×C9⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C92 | C32 |
| commutator | C3 | C3×C9×C27 |
| 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 | |
440 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);(11,13,16)(14,18,17)(19,20,23,21,24,29,26,31,36,22,25,30,27,32,37,34,39,42,28,33,38,35,40,43,41,44,45);(10,11,14,12,13,17,15,16,18)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4124);