Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92×C43 · Schur multiplier (H2) = C32×C9 |
| direct product | C9×C43×C9⋊C9 |
| unambiguous semidirect product | C9⋊(C92×C43) |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9×C43 | C32 |
| commutator | C3 | C3×C92×C43 |
| Frattini, cosocle | C33 | C33×C43 |
| socle, prime radical | C33×C43 | C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9×C43 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 31347 |
A minimal faithful permutation representation — 3 generators on 70 points:
(1,2,4,3,5,7,6,8,9)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(11,13,16)(14,18,17)(19,20,22,21,23,25,24,26,27);(10,11,14,12,13,17,15,16,18)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 106);