Finite Group DatabasebetaFinite Group Databasebeta| order | 37522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D18761 |
| unambiguous semidirect product | C73⋊D257 |
| unambiguous semidirect product | C257⋊D73 |
| Fitting, socle, derived, Hall | C73×C257 ⋊ C2 |
| Sylow | C257 ⋊ D73 |
| Sylow | C73 ⋊ D257 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C73×C257 | C2 |
| LCS | G ▷ C73×C257 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 330 |
| transitive degree | μt(G) = 18761 |
A minimal faithful permutation representation — 2 generators on 330 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)(74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330);(2,73)(3,72)(4,71)(5,70)(6,69)(7,68)(8,67)(9,66)(10,65)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(17,58)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(75,330)(76,329)(77,328)(78,327)(79,326)(80,325)(81,324)(82,323)(83,322)(84,321)(85,320)(86,319)(87,318)(88,317)(89,316)(90,315)(91,314)(92,313)(93,312)(94,311)(95,310)(96,309)(97,308)(98,307)(99,306)(100,305)(101,304)(102,303)(103,302)(104,301)(105,300)(106,299)(107,298)(108,297)(109,296)(110,295)(111,294)(112,293)(113,292)(114,291)(115,290)(116,289)(117,288)(118,287)(119,286)(120,285)(121,284)(122,283)(123,282)(124,281)(125,280)(126,279)(127,278)(128,277)(129,276)(130,275)(131,274)(132,273)(133,272)(134,271)(135,270)(136,269)(137,268)(138,267)(139,266)(140,265)(141,264)(142,263)(143,262)(144,261)(145,260)(146,259)(147,258)(148,257)(149,256)(150,255)(151,254)(152,253)(153,252)(154,251)(155,250)(156,249)(157,248)(158,247)(159,246)(160,245)(161,244)(162,243)(163,242)(164,241)(165,240)(166,239)(167,238)(168,237)(169,236)(170,235)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225)(181,224)(182,223)(183,222)(184,221)(185,220)(186,219)(187,218)(188,217)(189,216)(190,215)(191,214)(192,213)(193,212)(194,211)(195,210)(196,209)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)Polycyclic presentation:
2,73,257|b^a=b^72,c^a=c^2569382 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(37522, 3);