Finite Group DatabasebetaFinite Group Databasebeta| order | 38637 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C53 · Schur multiplier (H2) = C3 |
| direct product | C53×#10624460 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C53 | H(3,𝔽3) |
| commutator | C33 | C3×C9×C53 |
| Frattini, cosocle | C32×C9 | C32×C53 |
| socle | C32×C53 | C32⋊C9 |
| prime radical | C33×C53 | C3×C9 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C53 ◁ C32×C9×C53 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 12879 |
A minimal faithful permutation representation — 2 generators on 161 points:
(1,2,5)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21)(28,29,33,31,35,46,42,52,68,44,54,70,65,78,92,88,98,105,32,36,47,43,53,69,64,77,91)(30,37,55,40,57,80,62,82,100,39,51,45,61,76,67,86,97,90,41,34,48,63,50,72,87,75,94)(38,49,66,59,74,89,84,96,104,60,56,71,85,81,93,103,101,106,58,73,79,83,95,99,102,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161);(1,3,10,4,11,21,12,22,27)(2,6,16,8,17,20,7,9,19)(5,13,23,14,15,24,18,25,26)(28,30,38)(29,34,49)(31,40,59)(32,41,58)(33,45,66)(35,50,74)(36,51,73)(37,56,54)(39,60,44)(42,62,84)(43,63,83)(46,67,89)(47,55,79)(48,71,70)(52,75,96)(53,76,95)(57,81,78)(61,85,65)(64,87,102)(68,90,104)(69,80,99)(72,93,92)(77,97,107)(82,101,98)(86,103,88)(91,100,108)(94,106,105)5565 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38637, 20);