Finite Group DatabasebetaFinite Group Databasebeta| order | 37682 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D18841 |
| unambiguous semidirect product | C83⋊D227 |
| unambiguous semidirect product | C227⋊D83 |
| Fitting, socle, derived, Hall | C83×C227 ⋊ C2 |
| Sylow | C227 ⋊ D83 |
| Sylow | C83 ⋊ D227 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C83×C227 | C2 |
| LCS | G ▷ C83×C227 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 310 |
| transitive degree | μt(G) = 18841 |
A minimal faithful permutation representation — 2 generators on 310 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83)(84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310);(2,83)(3,82)(4,81)(5,80)(6,79)(7,78)(8,77)(9,76)(10,75)(11,74)(12,73)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,55)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(85,310)(86,309)(87,308)(88,307)(89,306)(90,305)(91,304)(92,303)(93,302)(94,301)(95,300)(96,299)(97,298)(98,297)(99,296)(100,295)(101,294)(102,293)(103,292)(104,291)(105,290)(106,289)(107,288)(108,287)(109,286)(110,285)(111,284)(112,283)(113,282)(114,281)(115,280)(116,279)(117,278)(118,277)(119,276)(120,275)(121,274)(122,273)(123,272)(124,271)(125,270)(126,269)(127,268)(128,267)(129,266)(130,265)(131,264)(132,263)(133,262)(134,261)(135,260)(136,259)(137,258)(138,257)(139,256)(140,255)(141,254)(142,253)(143,252)(144,251)(145,250)(146,249)(147,248)(148,247)(149,246)(150,245)(151,244)(152,243)(153,242)(154,241)(155,240)(156,239)(157,238)(158,237)(159,236)(160,235)(161,234)(162,233)(163,232)(164,231)(165,230)(166,229)(167,228)(168,227)(169,226)(170,225)(171,224)(172,223)(173,222)(174,221)(175,220)(176,219)(177,218)(178,217)(179,216)(180,215)(181,214)(182,213)(183,212)(184,211)(185,210)(186,209)(187,208)(188,207)(189,206)(190,205)(191,204)(192,203)(193,202)(194,201)(195,200)(196,199)(197,198)Polycyclic presentation:
2,83,227|b^a=b^82,c^a=c^2269422 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37682, 3);