Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C92 · Schur multiplier (H2) = C34×C9 |
| direct product | C3×C9×C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C92 | C32 |
| commutator | C3 | C32×C92 |
| Frattini, cosocle | C3×C9 | C34 |
| socle | C33 | C32×C9 |
| prime radical | C34 | C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C92 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3);(4,5,6,7,8,9,10,11,12);(13,14,17,15,19,25,21,28,33,16,20,24,22,29,32,30,36,38,23,18,26,31,27,34,37,35,39);(14,18,20)(17,24,26)(19,27,29)(25,32,34)(28,35,36)(33,38,39)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 7959);