Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624530 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C43 | #732 |
| commutator | C3×C9 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| socle | C3×C43 | H(3,𝔽3)⋊C9 |
| prime radical | C43×#6908 | C3 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C43 ◁ C3×C9×C43 ◁ C32×C9×C43 ◁ G | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,8,5,11,28,22,35,58,7,12,29,23,36,59,21,10,27,6,24,51,20,49,78,4,18,47)(3,13,37,16,33,62,15,39,26,17,34,55,45,52,61,46,68,57,25,48,70,50,9,30,19,32,56)(14,40,73,42,67,54,44,65,79,43,74,81,76,72,80,77,31,63,53,64,60,75,38,71,41,66,69)(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);(1,3,14)(2,9,31)(4,19,41)(5,16,42)(6,25,53)(7,17,43)(8,26,54)(10,34,67)(11,32,64)(12,33,66)(13,38,35)(15,44,22)(18,48,72)(20,50,75)(21,46,77)(23,45,76)(24,52,65)(27,30,60)(28,55,79)(29,57,80)(36,39,40)(37,69,78)(47,62,73)(49,68,74)(51,56,71)(58,61,81)(59,70,63)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 90);