Polytope of Type {2,81,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,81,6}*1944
if this polytope has a name.
Group : SmallGroup(1944,955)
Rank : 4
Schlafli Type : {2,81,6}
Number of vertices, edges, etc : 2, 81, 243, 6
Order of s₀s₁s₂s₃ : 162
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,81,2}*648, {2,27,6}*648
   9-fold quotients : {2,27,2}*216, {2,9,6}*216
   27-fold quotients : {2,9,2}*72, {2,3,6}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6, 11)(  7, 10)(  8,  9)( 12, 29)( 13, 28)( 14, 27)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 30, 57)( 31, 59)( 32, 58)
( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 83)( 40, 82)
( 41, 81)( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)
( 49, 73)( 50, 72)( 51, 71)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 84,191)( 85,190)( 86,189)( 87,188)( 88,187)( 89,186)( 90,185)( 91,184)
( 92,183)( 93,182)( 94,181)( 95,180)( 96,179)( 97,178)( 98,177)( 99,176)
(100,175)(101,174)(102,173)(103,172)(104,171)(105,170)(106,169)(107,168)
(108,167)(109,166)(110,165)(111,245)(112,244)(113,243)(114,242)(115,241)
(116,240)(117,239)(118,238)(119,237)(120,236)(121,235)(122,234)(123,233)
(124,232)(125,231)(126,230)(127,229)(128,228)(129,227)(130,226)(131,225)
(132,224)(133,223)(134,222)(135,221)(136,220)(137,219)(138,218)(139,217)
(140,216)(141,215)(142,214)(143,213)(144,212)(145,211)(146,210)(147,209)
(148,208)(149,207)(150,206)(151,205)(152,204)(153,203)(154,202)(155,201)
(156,200)(157,199)(158,198)(159,197)(160,196)(161,195)(162,194)(163,193)
(164,192);;
s2 := (  3,111)(  4,113)(  5,112)(  6,119)(  7,118)(  8,117)(  9,116)( 10,115)
( 11,114)( 12,137)( 13,136)( 14,135)( 15,134)( 16,133)( 17,132)( 18,131)
( 19,130)( 20,129)( 21,128)( 22,127)( 23,126)( 24,125)( 25,124)( 26,123)
( 27,122)( 28,121)( 29,120)( 30, 84)( 31, 86)( 32, 85)( 33, 92)( 34, 91)
( 35, 90)( 36, 89)( 37, 88)( 38, 87)( 39,110)( 40,109)( 41,108)( 42,107)
( 43,106)( 44,105)( 45,104)( 46,103)( 47,102)( 48,101)( 49,100)( 50, 99)
( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)( 56, 93)( 57,138)( 58,140)
( 59,139)( 60,146)( 61,145)( 62,144)( 63,143)( 64,142)( 65,141)( 66,164)
( 67,163)( 68,162)( 69,161)( 70,160)( 71,159)( 72,158)( 73,157)( 74,156)
( 75,155)( 76,154)( 77,153)( 78,152)( 79,151)( 80,150)( 81,149)( 82,148)
( 83,147)(165,218)(166,217)(167,216)(168,215)(169,214)(170,213)(171,212)
(172,211)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)(179,204)
(180,203)(181,202)(182,201)(183,200)(184,199)(185,198)(186,197)(187,196)
(188,195)(189,194)(190,193)(191,192)(219,245)(220,244)(221,243)(222,242)
(223,241)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)
(231,233);;
s3 := ( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)( 36, 63)( 37, 64)
( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)( 44, 71)( 45, 72)
( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)( 52, 79)( 53, 80)
( 54, 81)( 55, 82)( 56, 83)(111,138)(112,139)(113,140)(114,141)(115,142)
(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)
(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)
(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(192,219)(193,220)
(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)(200,227)(201,228)
(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)(208,235)(209,236)
(210,237)(211,238)(212,239)(213,240)(214,241)(215,242)(216,243)(217,244)
(218,245);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  6, 11)(  7, 10)(  8,  9)( 12, 29)( 13, 28)( 14, 27)
( 15, 26)( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 30, 57)( 31, 59)
( 32, 58)( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 83)
( 40, 82)( 41, 81)( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)
( 48, 74)( 49, 73)( 50, 72)( 51, 71)( 52, 70)( 53, 69)( 54, 68)( 55, 67)
( 56, 66)( 84,191)( 85,190)( 86,189)( 87,188)( 88,187)( 89,186)( 90,185)
( 91,184)( 92,183)( 93,182)( 94,181)( 95,180)( 96,179)( 97,178)( 98,177)
( 99,176)(100,175)(101,174)(102,173)(103,172)(104,171)(105,170)(106,169)
(107,168)(108,167)(109,166)(110,165)(111,245)(112,244)(113,243)(114,242)
(115,241)(116,240)(117,239)(118,238)(119,237)(120,236)(121,235)(122,234)
(123,233)(124,232)(125,231)(126,230)(127,229)(128,228)(129,227)(130,226)
(131,225)(132,224)(133,223)(134,222)(135,221)(136,220)(137,219)(138,218)
(139,217)(140,216)(141,215)(142,214)(143,213)(144,212)(145,211)(146,210)
(147,209)(148,208)(149,207)(150,206)(151,205)(152,204)(153,203)(154,202)
(155,201)(156,200)(157,199)(158,198)(159,197)(160,196)(161,195)(162,194)
(163,193)(164,192);
s2 := Sym(245)!(  3,111)(  4,113)(  5,112)(  6,119)(  7,118)(  8,117)(  9,116)
( 10,115)( 11,114)( 12,137)( 13,136)( 14,135)( 15,134)( 16,133)( 17,132)
( 18,131)( 19,130)( 20,129)( 21,128)( 22,127)( 23,126)( 24,125)( 25,124)
( 26,123)( 27,122)( 28,121)( 29,120)( 30, 84)( 31, 86)( 32, 85)( 33, 92)
( 34, 91)( 35, 90)( 36, 89)( 37, 88)( 38, 87)( 39,110)( 40,109)( 41,108)
( 42,107)( 43,106)( 44,105)( 45,104)( 46,103)( 47,102)( 48,101)( 49,100)
( 50, 99)( 51, 98)( 52, 97)( 53, 96)( 54, 95)( 55, 94)( 56, 93)( 57,138)
( 58,140)( 59,139)( 60,146)( 61,145)( 62,144)( 63,143)( 64,142)( 65,141)
( 66,164)( 67,163)( 68,162)( 69,161)( 70,160)( 71,159)( 72,158)( 73,157)
( 74,156)( 75,155)( 76,154)( 77,153)( 78,152)( 79,151)( 80,150)( 81,149)
( 82,148)( 83,147)(165,218)(166,217)(167,216)(168,215)(169,214)(170,213)
(171,212)(172,211)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)
(179,204)(180,203)(181,202)(182,201)(183,200)(184,199)(185,198)(186,197)
(187,196)(188,195)(189,194)(190,193)(191,192)(219,245)(220,244)(221,243)
(222,242)(223,241)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)
(230,234)(231,233);
s3 := Sym(245)!( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)( 36, 63)
( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)( 44, 71)
( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)( 52, 79)
( 53, 80)( 54, 81)( 55, 82)( 56, 83)(111,138)(112,139)(113,140)(114,141)
(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)
(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)
(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(192,219)
(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)(200,227)
(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)(208,235)
(209,236)(210,237)(211,238)(212,239)(213,240)(214,241)(215,242)(216,243)
(217,244)(218,245);
poly := sub<Sym(245)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope