Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C37 |
| direct product | H(3,𝔽3)×C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C33 | C34 |
| commutator | C32 | C33×C9 |
| prime radical | C32×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
1619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,4)(3,6,8)(5,9,7);(2,5,6)(4,8,9);(11,13,16)(14,18,17)(19,20,22,21,23,25,24,26,27);(10,11,14,12,13,17,15,16,18)363 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 5915);