Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×#10624849 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C53 | C3×#732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C53 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×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)(3,13,41)(4,9,26)(5,21,27)(6,12,28)(8,29,18)(10,37,19)(11,40,24)(14,44,77)(15,32,70)(16,49,25)(17,38,58)(20,36,61)(22,31,59)(23,35,63)(30,65,50)(33,62,53)(34,71,54)(39,73,55)(42,76,45)(43,60,51)(46,67,75)(47,69,64)(48,74,57)(52,66,79)(56,68,80)(72,81,78)(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)(2,8,30,12,33,68,11,32,67)(4,18,52)(5,22,56)(6,17,47)(7,25,57,40,71,76,28,59,79)(9,34,72,36,49,44,35,31,69)(10,38,74,39,43,66,21,13,42)(15,48,23)(16,50,55)(19,54,46)(20,53,45)(24,51,78)(26,60,80,63,58,65,61,41,75)(27,62,81,73,29,64,37,70,77);(1,4,19,6,20,55,24,23,5)(2,9,10,12,36,39,11,35,21)(3,15,16,17,18,22,51,53,54)(7,26,37,28,61,73,40,63,27)(8,31,43,33,34,13,32,49,38)(14,45,56,47,48,46,78,52,50)(25,58,29,59,60,62,71,41,70)(30,44,42,68,69,74,67,72,66)(57,75,81,79,65,77,76,80,64)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 409);