Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624462 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C32×C43 | C9⋊C9 |
| commutator | C9 | C3×C27×C43 |
| Frattini, cosocle | C92 | C32×C43 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C43 ◁ C92×C43 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 31347 |
A minimal faithful permutation representation — 2 generators on 97 points:
(2,5,11,7,14,23,18,26,13)(3,8,19)(6,15,12,16,27,24,17,22,25)(9,21,20)(28,29,32,30,33,38,35,40,45,31,34,39,36,41,46,43,48,51,37,42,47,44,49,52,50,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,6,3,5,12,9,13,15,4,7,17,8,14,25,20,11,22,10,18,16,19,26,24,21,23,27)4515 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 22);