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) = C37 |
| direct product | C32×#6868 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | H(3,𝔽3) |
| commutator | C32 | C33×C9 |
| Frattini, cosocle | C33 | C34 |
| prime radical | C35 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ C35 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
2600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2,3);(4,5,6);(7,8,10,9,11,13,12,14,15)(16,17,20)(18,24,28)(19,23,29)(21,30,26)(22,33,27)(25,35,38)(31,41,42)(32,40,37)(34,39,36);(16,18,25,27,37,42,19,26,36)(17,21,31,22,24,34,23,32,35)(20,28,38,33,40,41,29,30,39)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 7970);