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Dic₁₂₀₀₂ #72280418

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

Aliases

Splittings

Subgroups

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 378 points:

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

2^3,17,353|a^2=c,b^2=c,b^a=bc,d^a=d^16,e^a=e^352

Conjugacy classes & character table

12005 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 17, 353]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^352);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);