Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C43×C81 · Schur multiplier (H2) = C1 |
| direct product | C43×C243⋊C3 |
| unambiguous semidirect product | C243⋊(C3×C43) |
| subgroup | G/H | |
|---|---|---|
| center | C43×C81 | C32 |
| commutator | C3 | C3×C43×C81 |
| Frattini, cosocle | C81 | C32×C43 |
| socle | C3×C43 | C3×C81 |
| prime radical | C32×C43 | C81 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C43×C81 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 286 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 2 generators on 286 points:
(1,2,7,3,9,24,13,31,59,4,10,25,14,32,60,40,74,112,17,35,62,43,77,114,87,132,168,5,11,26,15,33,61,41,75,113,18,36,63,44,78,115,88,133,169,49,82,117,93,137,171,145,187,212,20,38,64,46,80,116,90,135,170,51,84,118,95,139,172,147,189,213,100,142,173,152,192,214,196,224,235,6,12,23,16,34,56,42,76,106,19,37,57,45,79,107,89,134,161,50,83,109,94,138,163,146,188,206,21,39,58,47,81,108,91,136,162,52,85,110,96,140,164,148,190,207,101,143,166,153,193,209,197,225,232,54,86,111,98,141,165,150,191,208,103,144,167,155,194,210,199,226,233,158,195,211,202,227,234,228,240,242,22,8,27,48,28,65,92,68,119,53,29,66,97,69,120,149,125,174,102,71,122,154,127,176,198,181,215,55,30,67,99,70,121,151,126,175,104,72,123,156,128,177,200,182,216,159,130,179,203,184,218,229,221,236,105,73,124,157,129,178,201,183,217,160,131,180,204,185,219,230,222,237,205,186,220,231,223,238,241,239,243)(244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286);(2,8,12)(7,23,27)(9,28,34)(10,29,37)(11,30,39)(24,56,65)(25,57,66)(26,58,67)(31,68,76)(32,69,79)(33,70,81)(35,71,83)(36,72,85)(38,73,86)(59,106,119)(60,107,120)(61,108,121)(62,109,122)(63,110,123)(64,111,124)(74,125,134)(75,126,136)(77,127,138)(78,128,140)(80,129,141)(82,130,143)(84,131,144)(112,161,174)(113,162,175)(114,163,176)(115,164,177)(116,165,178)(117,166,179)(118,167,180)(132,181,188)(133,182,190)(135,183,191)(137,184,193)(139,185,194)(142,186,195)(168,206,215)(169,207,216)(170,208,217)(171,209,218)(172,210,219)(173,211,220)(187,221,225)(189,222,226)(192,223,227)(212,232,236)(213,233,237)(214,234,238)(224,239,240)(235,242,243)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 94);