Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C27×C43 · Schur multiplier (H2) = C32 |
| direct product | C3×C43×C81⋊C3 |
| unambiguous semidirect product | C81⋊(C32×C43) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C27×C43 | C32 |
| commutator | C3 | C32×C27×C43 |
| Frattini, cosocle | C27 | C33×C43 |
| socle | C32×C43 | C3×C27 |
| prime radical | C33×C43 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27×C43 ◁ G | |
164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2,3)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(4,5,9,6,11,21,14,26,40,7,12,22,15,27,41,31,48,62,17,29,42,33,50,63,53,69,77,8,13,20,16,28,38,32,49,59,18,30,39,34,51,60,54,70,75,36,52,61,56,71,76,72,81,83,19,10,23,35,24,43,55,45,64,37,25,44,57,46,65,73,67,78,58,47,66,74,68,79,82,80,84);(5,10,13)(9,20,23)(11,24,28)(12,25,30)(21,38,43)(22,39,44)(26,45,49)(27,46,51)(29,47,52)(40,59,64)(41,60,65)(42,61,66)(48,67,70)(50,68,71)(62,75,78)(63,76,79)(69,80,81)(77,83,84)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 392);