Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C36 |
| direct product | C32×#6869 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C3×C9 | C34 |
| socle | C33 | C32⋊C9 |
| prime radical | C35 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3);(4,5,6);(7,8,11,9,14,19,15,23,27,10,12,18,16,20,26,24,28,32,17,13,21,25,22,29,31,30,33);(8,12,13)(14,20,22)(23,28,30)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 7977);