Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624456 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C43 | H(3,𝔽3) |
| commutator | C33 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| socle | C32×C43 | C32⋊C9 |
| prime radical | C3×H(3,𝔽3)×C43 | C9 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C43 ◁ C32×C9×C43 ◁ G | |
208 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)(3,5,8)(6,9,7)(10,11,15,13,17,28,24,34,50,14,18,29,25,35,51,46,59,73,26,36,52,47,60,74,70,80,87)(12,19,37,22,39,62,44,64,82,23,16,30,45,32,54,69,57,76,21,33,27,43,58,49,68,79,72)(20,31,48,41,56,71,66,78,86,40,55,61,65,77,81,84,89,90,42,38,53,67,63,75,85,83,88)(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)(10,12,20)(11,16,31)(13,22,41)(14,23,40)(15,27,48)(17,32,56)(18,33,55)(19,38,36)(21,42,26)(24,44,66)(25,45,65)(28,49,71)(29,37,61)(30,53,52)(34,57,78)(35,58,77)(39,63,60)(43,67,47)(46,69,84)(50,72,86)(51,62,81)(54,75,74)(59,79,89)(64,83,80)(68,85,70)(73,82,90)(76,88,87)4515 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 16);