Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×#10624840 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C53 | C3×#732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C53 |
| prime radical | C33×C53 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C53 ◁ C33×C53 ◁ C32×C9×C53 ◁ G | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 134 |
| transitive degree | μt(G) = 4293 |
A minimal faithful permutation representation — 3 generators on 134 points:
(1,2,6,5,10,22,19,9,30)(3,11,31,15,35,64,13,33,55)(4,16,21,18,8,27,47,29,61)(7,23,52,26,56,74,25,54,42)(12,36,68,37,69,79,39,70,53)(14,41,20,43,34,50,40,67,49)(17,45,51,46,44,60,75,28,62)(24,57,38,58,63,71,59,80,73)(32,48,76,65,77,81,66,78,72)(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);(1,3,12,5,15,37,19,13,39)(2,7,24,9,25,59,10,26,58)(4,14,38,18,43,71,47,40,73)(6,20,48)(8,11,32,16,33,66,29,35,65)(17,42,72,46,52,76,75,74,81)(21,23,53)(22,50,77)(27,56,68)(28,34,36,44,41,70,45,67,69)(30,49,78)(31,63,60)(51,55,57)(54,79,61)(62,64,80);(3,13,15)(7,25,26)(11,33,35)(12,37,39)(14,40,43)(20,49,50)(23,54,56)(24,58,59)(31,55,64)(32,65,66)(34,41,67)(36,69,70)(38,71,73)(42,74,52)(48,77,78)(53,68,79)(57,63,80)(72,76,81)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 400);