Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C43 · Schur multiplier (H2) = C9 |
| direct product | C43×#10624539 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6879 |
| commutator, Frattini, cosocle | C92 | C32×C43 |
| prime radical | C43×#6879 | C3 |
| LCS | G ▷ C92 ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C32×C43 ◁ C3×C9×C43 ◁ C92×C43 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,6)(3,11,34)(4,9,20)(5,10,23)(7,24,46)(8,28,39)(12,38,48)(13,42,19)(14,37,49)(15,29,50)(16,32,52)(17,30,21)(18,33,22)(25,58,72)(26,62,69)(27,51,44)(31,54,70)(35,55,73)(36,56,45)(40,68,66)(41,64,77)(43,60,76)(47,65,53)(57,74,63)(59,75,71)(61,78,80)(67,79,81)(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);(1,3,12,39,69,80,17,46,71,5,15,40,70,74,81,47,44,73,18,13,43,16,45,72,4,14,41)(2,7,25,32,11,35,65,36,68,10,27,61,8,29,64,9,26,60,33,37,67,31,42,59,30,63,38)(6,19,48,28,56,78,21,49,75,23,34,66,54,62,79,53,24,55,22,50,76,52,57,58,20,51,77)1763 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 99);