Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C53 · Schur multiplier (H2) = C33 |
| direct product | C53×#10624847 |
| 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:
(1,2,6,5,10,23,20,9,32)(3,11,33,15,29,66,13,35,59)(4,16,22,19,8,28,51,31,65)(7,24,47,27,55,49,26,58,44)(12,37,73,38,72,57,40,74,80)(14,42,21,17,36,54,41,68,53)(18,48,56,50,45,64,78,30,69)(25,60,46,61,81,39,63,70,43)(34,71,76,67,52,77,62,79,75)(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)(2,7,25,10,27,61,9,26,63)(4,17,46)(5,15,38)(6,21,52,32,53,71,23,54,79)(8,29,67,31,35,62,16,11,34)(13,40,20)(14,43,51)(18,49,76)(19,41,39)(22,55,80,65,24,57,28,58,73)(30,42,37,48,36,72,45,68,74)(33,70,64,59,81,56,66,60,69)(44,77,50)(47,75,78);(3,13,15)(7,26,27)(11,35,29)(12,38,40)(14,41,17)(21,53,54)(24,58,55)(25,61,63)(33,59,66)(34,67,62)(36,42,68)(37,72,74)(39,43,46)(44,49,47)(52,79,71)(57,80,73)(60,81,70)(75,76,77)3021 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 407);