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#72234404

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

Aliases

No aliases for this group.

Subgroups

1131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 3 generators on 272 points:

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

Polycyclic presentation:

2^12|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,g^2=j,h^2=kl,i^2=l,j^2=k,k^2=l,c^a=chk,c^b=cl,d^c=dkl,f^a=fk,f^d=fl,g^c=gl,i^a=il

Conjugacy classes & character table

736 conjugacy classes — view the full character table.

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");;
gap> coll := SingleCollector(F, [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*F.12);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.11);;
gap> SetPower(coll, 11, F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.12);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);