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#72234308

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial

Aliases

No aliases for this group.

Subgroups

1808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 288 points:

(1,2,3,7,4,8,11,17,5,9,12,18,14,20,23,27,6,10,13,19,15,21,24,28,16,22,25,29,26,30,31,32)(33,34,37,44)(35,42,50,73)(36,55,56,113,58,115,117,168)(38,45,62,85)(39,47,64,87)(40,46,65,89)(41,48,63,88)(43,78,79,121,82,157,160,59)(49,96,97,180,99,182,184,233)(51,74,103,145)(52,76,105,147)(53,75,106,149)(54,77,104,148)(57,114,116,205,118,207,208,251)(60,83,61,124,122,165,126,206)(66,91,129,170)(67,90,130,172)(68,92,128,171)(69,93,133,175)(70,95,131,174)(71,94,132,86)(72,138,139,188,142,222,225,100)(80,156,158,209,162,245,247,119)(81,127,84,166,164,215,159,123)(98,181,183,256,185,258,259,281)(101,143,102,191,189,230,193,257)(107,151,196,235)(108,150,197,237)(109,152,195,236)(110,153,200,240)(111,155,198,239)(112,154,199,146)(120,163,125,212,210,249,213,272)(134,176,218,254)(135,178,216,253)(136,177,217,169)(137,179,219,173)(140,221,223,260,227,275,277,186)(141,194,144,231,229,266,224,190)(161,214,167,250,248,273,246,211)(187,228,192,263,261,279,264,286)(201,241,269,284)(202,243,267,283)(203,242,268,234)(204,244,270,238)(220,255,274,252)(226,265,232,280,278,287,276,262)(271,285,288,282);(33,35,38,51,39,52,66,107,40,53,67,108,69,110,134,201,41,54,68,109,70,111,135,202,71,112,136,203,137,204,220,271)(34,42,45,74,47,76,91,151,46,75,90,150,93,153,176,241,48,77,92,152,95,155,178,243,94,154,177,242,179,244,255,285)(36,49,57,98,59,100,119,186,60,101,120,187,123,190,211,262,58,99,118,185,121,188,209,260,122,189,210,261,166,231,250,280)(37,50,62,103,64,105,129,196,65,106,130,197,133,200,218,269,63,104,128,195,131,198,216,267,132,199,217,268,219,270,274,288)(43,72,80,140,83,143,163,228,81,141,161,226,115,182,207,258,82,142,162,227,165,230,249,279,164,229,248,278,55,96,114,181)(44,73,85,145,87,147,170,235,89,149,172,237,175,240,254,284,88,148,171,236,174,239,253,283,86,146,169,234,173,238,252,282)(56,97,116,183,78,138,156,221,61,102,125,192,127,194,214,265,117,184,208,259,157,222,245,275,126,193,213,264,215,266,273,287)(79,139,158,223,124,191,212,263,84,144,167,232,168,233,251,281,160,225,247,277,206,257,272,286,159,224,246,276,113,180,205,256);(33,36)(34,43,46,81,48,82,94,164)(35,49)(37,61,63,126)(38,57)(39,59)(40,60)(41,58)(42,72,75,141,77,142,154,229)(44,84,86,79,88,159,89,160)(45,80,90,161,92,162,177,248)(47,83,93,115,95,165,179,55)(50,102,104,193)(51,98)(52,100)(53,101)(54,99)(56,65,117,132)(62,125,128,213)(64,127,131,215)(66,119)(67,120)(68,118)(69,123)(70,121)(71,122)(73,144,146,139,148,224,149,225)(74,140,150,226,152,227,242,278)(76,143,153,182,155,230,244,96)(78,133,157,219)(85,167,169,158,171,246,172,247)(87,168,173,124,174,113,175,206)(91,163,176,207,178,249,255,114)(97,106,184,199)(103,192,195,264)(105,194,198,266)(107,186)(108,187)(109,185)(110,190)(111,188)(112,189)(116,130,208,217)(129,214,216,273)(134,211)(135,209)(136,210)(137,166)(138,200,222,270)(145,232,234,223,236,276,237,277)(147,233,238,191,239,180,240,257)(151,228,241,258,243,279,285,181)(156,218,245,274)(170,251,252,212,253,205,254,272)(183,197,259,268)(196,265,267,287)(201,262)(202,260)(203,261)(204,231)(220,250)(221,269,275,288)(235,281,282,263,283,256,284,286)(271,280)

Polycyclic presentation:

2^13|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=j,h^2=k,j^2=l,k^2=m,c^a=chkm,d^c=dk,f^a=fkm,f^d=fm,g^c=gm,i^a=im

Conjugacy classes & character table

1472 conjugacy classes. The character table is too large to display.

Relationships

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 10, F.12);;
gap> SetPower(coll, 11, F.13);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.11*F.13);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11*F.13);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
gap> SetConjugate(coll, 7, 3, F.7*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);