Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C3 |
| direct product | C53×#10624830 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C53 | #6895 |
| commutator, Frattini, cosocle | C3×C9 | C33×C53 |
| prime radical | C33×C53 | 3−1+2 |
| LCS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C53 ◁ C3×C9×C53 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 134 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 3 generators on 134 points:
(1,2,6,4,10,23,17,31,50,5,11,22,18,34,51,46,30,52,19,9,32,16,33,21,45,8,29)(3,12,20,14,37,49,42,26,59,15,7,24,43,27,55,72,60,75,44,28,54,41,61,48,71,36,62)(13,38,47,39,66,74,68,63,78,40,35,53,69,64,76,81,57,77,70,25,56,67,58,73,80,65,79)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134);(1,3,13,19,44,70,5,15,40)(2,7,25,9,12,35,11,28,38)(4,14,39,16,41,67,18,43,69)(6,20,47,32,54,56,22,24,53)(8,26,57,30,36,63,31,60,65)(10,27,58,33,37,64,34,61,66)(17,42,68,45,71,80,46,72,81)(21,48,73,51,55,76,23,49,74)(29,62,79,52,75,77,50,59,78);(2,8,10,9,30,33,11,31,34)(4,16,18)(6,21,50,22,23,52,32,51,29)(7,26,27,12,36,37,28,60,61)(14,41,43)(17,46,45)(20,48,59,24,49,75,54,55,62)(25,57,58,35,63,64,38,65,66)(39,67,69)(42,72,71)(47,73,78,53,74,77,56,76,79)(68,81,80)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 390);