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#72234420

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

Aliases

No aliases for this group.

Subgroups

1995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 2 generators on 320 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)(65,66,68,75)(67,81,82,135,85,138,144,243,87,141,145,244,88,139,146,245)(69,95,96,180,98,182,171,220)(70,77,91,117)(71,79,93,119)(72,78,94,121)(73,80,92,120)(74,108,109,195,111,197,202,272,114,198,203,273,112,200,204,155)(76,122,123,189,125,230,221,99)(83,147,148,261,150,248,142,255,152,262,256,275,159,274,257,319)(84,137,143,215,154,213,258,216,156,212,259,217,157,113,199,115)(86,140,89,162,153,250,164,276,160,254,165,277,161,253,166,278)(90,167,158,236,169,281,100,127)(97,181,184,292,186,293,284,308)(101,128,172,222)(102,130,174,224)(103,129,175,226)(104,131,173,225)(105,132,178,228)(106,134,176,227)(107,133,177,118)(110,205,206,267,207,301,201,271,209,304,302,149,211,263,303,265)(116,185,210,260,219,190,126,170)(124,229,232,295,233,313,309,187)(136,183,246,268,247,163,242,269,249,279,316,264,252,280,317,151)(168,241,270,314,282,315,188,235)(179,289,290,320,208,196,231,298,291,299,214,297,305,300,306,266)(191,237,285,310)(192,238,288,312)(193,240,286,311)(194,239,287,223)(218,294,251,318,307,296,234,283);(33,34,35,39,36,40,43,49,37,41,44,50,46,52,55,59,38,42,45,51,47,53,56,60,48,54,57,61,58,62,63,64)(65,67,69,83)(66,74,76,110,78,112,126,207,80,114,125,209,133,111,210,211)(68,89,90,163,92,165,169,280)(70,84,97,149)(71,86,99,151)(72,88,100,150)(73,87,98,152)(75,115,116,214,118,215,123,231,120,216,219,290,121,217,221,306)(77,81,124,136,129,139,234,247,131,141,233,249,239,138,251,252)(79,113,127,208,132,212,182,291,134,213,236,305,128,137,95,179)(82,122,142,178,145,230,257,172)(85,158,159,107)(91,108,168,196,173,198,282,300)(93,146,170,256,176,144,260,148)(94,166,171,279,177,164,96,183)(101,153,185,264)(102,155,187,266)(103,157,188,265)(104,156,186,267)(105,161,190,268)(106,160,189,269)(109,167,201,227,202,180,206,228,203,281,303,119,204,220,302,222)(117,162,218,275,223,276,232,255,225,277,307,261,226,278,309,319)(130,140,235,248,238,253,293,262,240,254,314,274,237,250,181,147)(135,241,242,311,243,292,246,312,244,315,317,224,245,308,316,310)(143,229,205,288,259,313,304,285)(154,270,271,194)(174,199,283,301,286,258,318,263)(175,200,284,299,287,197,184,289)(191,195,294,297)(192,273,296,320)(193,272,295,298)

Polycyclic presentation:

2^14|a^2=c,b^2=d,c^2=f,d^2=g,e^2=hkn,f^2=i,g^2=j,h^2=k,i^2=l,j^2=m,k^2=n,b^a=be,c^b=ch,d^a=dhkn,d^c=dkn,f^b=fk,f^d=fn,g^a=gkn,g^c=gn,i^b=in,j^a=jn

Conjugacy classes & character table

1504 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", "n");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.8*F.11*F.14);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.12);;
gap> SetPower(coll, 10, F.13);;
gap> SetPower(coll, 11, F.14);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.11*F.14);;
gap> SetConjugate(coll, 4, 3, F.4*F.11*F.14);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.14);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.14);;
gap> SetConjugate(coll, 7, 3, F.7*F.14);;
gap> SetConjugate(coll, 9, 2, F.9*F.14);;
gap> SetConjugate(coll, 10, 1, F.10*F.14);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);