Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×#10624845 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C53 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C53 |
| socle | C3×C53 | C3×#732 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C53 ◁ C3×C9×C53 ◁ C32×C9×C53 ◁ G | |
188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 134 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 3 generators on 134 points:
(1,2,7)(3,13,39)(4,9,25)(5,20,26)(6,12,27)(8,28,15)(10,35,18)(11,38,23)(14,43,75)(16,50,24)(17,36,70)(19,34,58)(21,41,57)(22,33,60)(29,63,51)(30,62,47)(31,68,54)(32,59,49)(37,69,52)(40,73,76)(42,56,48)(44,65,80)(45,66,72)(46,74,61)(53,67,79)(55,78,71)(64,81,77)(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);(1,3,14)(2,8,29,12,32,67,11,31,66)(4,15,44)(5,21,53)(6,17,46)(7,24,55,38,62,73,27,57,80)(9,30,64,34,50,43,33,41,74)(10,36,71,37,42,65,20,13,40)(16,51,52)(18,47,45)(19,49,76)(22,54,78)(23,48,77)(25,56,79,60,70,63,58,39,72)(26,59,81,69,28,61,35,68,75);(1,4,18,6,19,52,23,22,5)(2,9,10,12,34,37,11,33,20)(3,15,47,17,49,16,48,54,21)(7,25,35,27,58,69,38,60,26)(8,30,36,32,50,42,31,41,13)(14,44,45,46,76,51,77,78,53)(24,56,68,57,39,28,62,70,59)(29,64,71,67,43,65,66,74,40)(55,79,75,80,72,61,73,63,81)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 405);