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Dic₈₆₀₈ #46809103

• 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 397 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);(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)(130,397)(131,396)(132,395)(133,394)(134,393)(135,392)(136,391)(137,390)(138,389)(139,388)(140,387)(141,386)(142,385)(143,384)(144,383)(145,382)(146,381)(147,380)(148,379)(149,378)(150,377)(151,376)(152,375)(153,374)(154,373)(155,372)(156,371)(157,370)(158,369)(159,368)(160,367)(161,366)(162,365)(163,364)(164,363)(165,362)(166,361)(167,360)(168,359)(169,358)(170,357)(171,356)(172,355)(173,354)(174,353)(175,352)(176,351)(177,350)(178,349)(179,348)(180,347)(181,346)(182,345)(183,344)(184,343)(185,342)(186,341)(187,340)(188,339)(189,338)(190,337)(191,336)(192,335)(193,334)(194,333)(195,332)(196,331)(197,330)(198,329)(199,328)(200,327)(201,326)(202,325)(203,324)(204,323)(205,322)(206,321)(207,320)(208,319)(209,318)(210,317)(211,316)(212,315)(213,314)(214,313)(215,312)(216,311)(217,310)(218,309)(219,308)(220,307)(221,306)(222,305)(223,304)(224,303)(225,302)(226,301)(227,300)(228,299)(229,298)(230,297)(231,296)(232,295)(233,294)(234,293)(235,292)(236,291)(237,290)(238,289)(239,288)(240,287)(241,286)(242,285)(243,284)(244,283)(245,282)(246,281)(247,280)(248,279)(249,278)(250,277)(251,276)(252,275)(253,274)(254,273)(255,272)(256,271)(257,270)(258,269)(259,268)(260,267)(261,266)(262,265)(263,264)

Polycyclic presentation:

2^7,269|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^268,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^268

Conjugacy classes & character table

8611 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34432, 2728);