Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624479 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6856 |
| 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 | |
110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,6,5,10,23,20,9,32)(3,11,33,15,36,68,14,29,58)(4,16,22,19,8,28,51,31,65)(7,24,49,27,59,47,26,55,42)(12,37,72,39,71,57,44,73,80)(13,40,21,43,35,53,17,46,54)(18,48,56,50,45,64,78,30,66)(25,60,76,63,81,38,62,67,41)(34,69,75,70,52,77,61,79,74)(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);(1,3,12,5,15,39,20,14,44)(2,7,25,10,27,63,9,26,62)(4,17,38,19,13,41,51,43,76)(6,21,52,23,53,79,32,54,69)(8,29,61,31,11,34,16,36,70)(18,49,74,50,47,75,78,42,77)(22,55,72,28,24,57,65,59,80)(30,46,73,48,40,37,45,35,71)(33,67,64,68,60,66,58,81,56)1763 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 39);