Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C35 · Schur multiplier (H2) = C38 |
| direct product | C32×#6899 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | C33 |
| commutator, Frattini, cosocle | C32 | C35 |
| prime radical | C35 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
4634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2,3);(4,5,6);(8,11,9)(12,13,14)(16,17,21)(18,25,37)(19,23,31)(20,24,32)(22,33,28)(26,36,40)(27,38,30)(29,35,41)(34,42,39);(7,8,12,15,9,13,10,11,14)(16,18,26,20,27,39,29,28,19)(17,22,34,24,25,23,35,38,36)(21,30,31,32,33,40,41,37,42);(7,9,14,15,11,12,10,8,13)(16,19,28,29,39,27,20,26,18)(17,23,22,35,34,38,24,36,25)(21,31,33,41,42,30,32,40,37)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9154);