Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C43×C81 · Schur multiplier (H2) = C32 |
| direct product | C43×C32⋊C81 |
| unambiguous semidirect product | C32⋊(C43×C81) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C27×C43 | C32 |
| commutator | C3 | C3×C43×C81 |
| Frattini, cosocle | C3×C27 | C32×C43 |
| socle | C32×C43 | C3×C27 |
| prime radical | C33×C43 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27×C43 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,15,12,16,25,19,28,41,13,17,26,20,29,42,34,47,60,22,31,44,36,49,62,54,67,76,14,18,27,21,30,43,35,48,61,23,32,45,37,50,63,55,68,77,39,52,65,57,70,79,73,82,86,24,33,46,38,51,64,56,69,78,40,53,66,58,71,80,74,83,87,59,72,81,75,84,88,85,89,90)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(2,5,6)(4,8,9)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 61);