Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C3×C53×#6878 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C53 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C53 |
| prime radical | C3×C53×#732 | C3 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C53 ◁ C33×C53 ◁ C32×C9×C53 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(4,5,8)(6,13,17)(7,12,18)(9,19,14)(10,23,15)(11,24,16)(20,28,26)(21,29,30)(22,27,25);(5,9,20,12,22,13,11,21,10)(8,17,19,24,28,29,18,23,27)5565 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 394);