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Dic₈₆₇₂ #46834582

• metacyclic
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

Splittings

Subgroups

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 399 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)(65,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399);(1,65,33,97)(2,66,34,98)(3,67,35,99)(4,68,36,100)(5,69,37,101)(6,70,38,102)(7,71,39,103)(8,72,40,104)(9,73,41,105)(10,74,42,106)(11,75,43,107)(12,76,44,108)(13,77,45,109)(14,78,46,110)(15,79,47,111)(16,80,48,112)(17,81,49,113)(18,82,50,114)(19,83,51,115)(20,84,52,116)(21,85,53,117)(22,86,54,118)(23,87,55,119)(24,88,56,120)(25,89,57,121)(26,90,58,122)(27,91,59,123)(28,92,60,124)(29,93,61,125)(30,94,62,126)(31,95,63,127)(32,96,64,128)(130,399)(131,398)(132,397)(133,396)(134,395)(135,394)(136,393)(137,392)(138,391)(139,390)(140,389)(141,388)(142,387)(143,386)(144,385)(145,384)(146,383)(147,382)(148,381)(149,380)(150,379)(151,378)(152,377)(153,376)(154,375)(155,374)(156,373)(157,372)(158,371)(159,370)(160,369)(161,368)(162,367)(163,366)(164,365)(165,364)(166,363)(167,362)(168,361)(169,360)(170,359)(171,358)(172,357)(173,356)(174,355)(175,354)(176,353)(177,352)(178,351)(179,350)(180,349)(181,348)(182,347)(183,346)(184,345)(185,344)(186,343)(187,342)(188,341)(189,340)(190,339)(191,338)(192,337)(193,336)(194,335)(195,334)(196,333)(197,332)(198,331)(199,330)(200,329)(201,328)(202,327)(203,326)(204,325)(205,324)(206,323)(207,322)(208,321)(209,320)(210,319)(211,318)(212,317)(213,316)(214,315)(215,314)(216,313)(217,312)(218,311)(219,310)(220,309)(221,308)(222,307)(223,306)(224,305)(225,304)(226,303)(227,302)(228,301)(229,300)(230,299)(231,298)(232,297)(233,296)(234,295)(235,294)(236,293)(237,292)(238,291)(239,290)(240,289)(241,288)(242,287)(243,286)(244,285)(245,284)(246,283)(247,282)(248,281)(249,280)(250,279)(251,278)(252,277)(253,276)(254,275)(255,274)(256,273)(257,272)(258,271)(259,270)(260,269)(261,268)(262,267)(263,266)(264,265)

Polycyclic presentation:

2^7,271|a^2=g,b^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^270,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^270

Conjugacy classes & character table

8675 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34688, 2728);