Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C43 · Schur multiplier (H2) = C1 |
| direct product | C43×#10624532 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C43 | C9⋊C9 |
| commutator | C9 | C3×C27×C43 |
| Frattini, cosocle | C3×C27 | C32×C43 |
| socle | C3×C43 | C9⋊C27 |
| prime radical | C32×C43 | C27⋊C3 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C43 ◁ C3×C27×C43 ◁ G | |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,8,5,11,32,24,43,64,6,12,31,25,44,63,59,42,74,27,46,62,18,50,78,51,75,81,7,13,33,26,45,30,58,76,60,28,10,39,54,9,34,15,36,66,20,41,72,56,37,67,23,35,69,29,47,65,16,53,61,55,70,79,4,19,57,21,38,68,52,71,80,22,40,73,3,14,48,17,49,77)(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);(1,3,15,4,16,51,20,54,24,7,18,52,6,5,23,22,21,58,29,56,59,28,26,17,27,25,55)(2,9,35,10,14,43,40,45,36,13,38,49,19,50,76,46,53,71,47,44,75,12,37,70,41,11,42)(8,30,60,31,34,66,62,67,69,33,61,79,39,68,80,72,48,77,65,32,64,57,63,74,73,78,81)4515 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 92);