Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×#10624790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C53 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C53 |
| prime radical | C3×C53×#732 | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C53 ◁ C32×C9×C53 ◁ G | |
256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 12879 |
A minimal faithful permutation representation — 3 generators on 89 points:
(2,4,7)(5,9,8)(10,11,14)(12,18,22)(13,17,23)(15,24,19)(16,27,21)(20,31,33)(25,34,29)(26,36,32)(28,35,30)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,5,3,4,8,6,7,9)(12,19,29)(15,25,18)(20,32,30)(22,24,34)(26,28,31)(33,36,35);(10,12,20,21,29,32,13,19,30)(11,15,26,16,18,28,17,25,31)(14,22,33,27,34,36,23,24,35)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 350);