Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C9 · Schur multiplier (H2) = C39 |
| direct product | C9×3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C34 |
| commutator | C3 | C34×C9 |
| Frattini, socle, cosocle | C32 | C35 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
5484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4,5,6,7,8,9);(10,11,14,13,17,24,22,16,28)(12,18,29,21,27,23,20,15,25)(19,31,35,32,26,36,33,30,34);(10,12,19)(11,15,26)(13,21,32)(14,23,34)(16,27,31)(17,18,30)(20,33,22)(24,25,35)(28,29,36);(12,20,21)(15,27,18)(19,32,33)(23,29,25)(26,30,31)(34,35,36);(11,16,17)(14,24,28)(15,27,18)(23,25,29)(26,31,30)(34,35,36)747 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9102);