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) = C310 |
| direct product | C32×C9⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | C33 |
| commutator, Frattini, cosocle | C32 | C35 |
| prime radical | C33×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
8594 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3);(4,5,6);(8,10,13)(11,15,14)(16,17,19)(18,21,23)(20,24,22);(7,8,11,9,10,14,12,13,15);(17,20,21)(19,23,24)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9113);