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D₂₇₇₉₇ #72291121

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

Aliases

Splittings

Subgroups

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 379 points:

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

Polycyclic presentation:

2,7,11,19^2|d^19=e,b^a=b^6,c^a=c^10,d^a=d^18e^18,e^a=e^18

Conjugacy classes & character table

13900 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, 7, 11, 19, 19]);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2^6);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^18*F.5^18);;
gap> SetConjugate(coll, 5, 1, F.5^18);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);