Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624523 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C43 | #732 |
| commutator | C3×C9 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| socle | C3×C43 | #6868 |
| prime radical | C33×C43 | 3−1+2 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C43 ◁ C3×C9×C43 ◁ C32×C9×C43 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,7,5,10,26,22,37,58,23,11,38,21,9,33,19,36,56,6,12,27,4,17,25,20,35,57)(3,13,39,15,42,24,47,8,28,48,30,60,51,65,69,52,31,66,16,34,67,18,41,54,50,32,55)(14,43,73,45,29,61,77,62,79,78,74,80,44,63,70,76,68,53,46,72,59,49,64,81,75,40,71)(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);(1,3,14)(2,8,29,12,32,64,11,31,63)(4,18,49)(5,15,45)(6,16,46)(7,24,53,38,69,71,27,54,79)(9,34,68,10,30,62,17,13,40)(19,52,76)(20,50,75)(21,51,44)(22,47,77)(23,48,78)(25,55,80,26,28,59,33,66,73)(35,42,43,36,41,72,37,65,74)(39,70,58,60,81,56,67,61,57)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 83);