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) = C3×C9 |
| direct product | C27×C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C27 | C32 |
| commutator | C3 | C3×C9×C27 |
| Frattini, cosocle | C92 | C33 |
| socle | C32 | C3×C92 |
| prime radical | C33 | C92 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C27 ◁ G | |
| LECS | G ▷ C92 ▷ C32 ▷ C1 | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 3 generators on 54 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(28,29,32,30,34,40,36,43,48,31,35,39,37,44,47,45,51,53,38,33,41,46,42,49,52,50,54);(29,33,35)(32,39,41)(34,42,44)(40,47,49)(43,50,51)(48,53,54)891 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 411);