Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624486 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6857 |
| commutator, Frattini, cosocle | C32×C9 | C32×C43 |
| prime radical | C33×C43 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C33×C43 ◁ C32×C9×C43 ◁ G | |
118 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)(3,11,32,14,35,66,15,27,55)(4,8,21)(5,10,22)(7,23,45,25,56,46,26,57,41)(9,31,19)(12,36,69,38,70,79,43,72,54)(13,39,50,16,44,51,42,34,20)(17,28,52)(18,30,53)(24,58,40,59,65,74,60,80,37)(29,64,48)(33,67,71,61,49,73,68,77,75)(47,63,78)(62,81,76)(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);(1,3,12,5,15,38,19,14,43)(2,7,24,10,26,59,9,25,60)(4,16,37,18,13,40,48,42,74)(6,20,49,22,51,77,31,50,67)(8,27,61,30,35,68,29,11,33)(17,46,71,47,45,73,76,41,75)(21,23,54,53,57,69,64,56,79)(28,39,72,63,34,36,62,44,70)(32,65,52,55,80,78,66,58,81)1763 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 46);