Polytope of Type {81,4,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {81,4,2}*1296
if this polytope has a name.
Group : SmallGroup(1296,630)
Rank : 4
Schlafli Type : {81,4,2}
Number of vertices, edges, etc : 81, 162, 4, 2
Order of s₀s₁s₂s₃ : 162
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {27,4,2}*432
   9-fold quotients : {9,4,2}*144
   27-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)( 43,106)
( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)( 51, 86)
( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)
( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)( 67, 82)
( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(109,313)(110,315)(111,314)
(112,316)(113,321)(114,323)(115,322)(116,324)(117,317)(118,319)(119,318)
(120,320)(121,301)(122,303)(123,302)(124,304)(125,309)(126,311)(127,310)
(128,312)(129,305)(130,307)(131,306)(132,308)(133,289)(134,291)(135,290)
(136,292)(137,297)(138,299)(139,298)(140,300)(141,293)(142,295)(143,294)
(144,296)(145,277)(146,279)(147,278)(148,280)(149,285)(150,287)(151,286)
(152,288)(153,281)(154,283)(155,282)(156,284)(157,265)(158,267)(159,266)
(160,268)(161,273)(162,275)(163,274)(164,276)(165,269)(166,271)(167,270)
(168,272)(169,253)(170,255)(171,254)(172,256)(173,261)(174,263)(175,262)
(176,264)(177,257)(178,259)(179,258)(180,260)(181,241)(182,243)(183,242)
(184,244)(185,249)(186,251)(187,250)(188,252)(189,245)(190,247)(191,246)
(192,248)(193,229)(194,231)(195,230)(196,232)(197,237)(198,239)(199,238)
(200,240)(201,233)(202,235)(203,234)(204,236)(205,217)(206,219)(207,218)
(208,220)(209,225)(210,227)(211,226)(212,228)(213,221)(214,223)(215,222)
(216,224);;
s1 := (  1,109)(  2,110)(  3,112)(  4,111)(  5,117)(  6,118)(  7,120)(  8,119)
(  9,113)( 10,114)( 11,116)( 12,115)( 13,137)( 14,138)( 15,140)( 16,139)
( 17,133)( 18,134)( 19,136)( 20,135)( 21,141)( 22,142)( 23,144)( 24,143)
( 25,125)( 26,126)( 27,128)( 28,127)( 29,121)( 30,122)( 31,124)( 32,123)
( 33,129)( 34,130)( 35,132)( 36,131)( 37,205)( 38,206)( 39,208)( 40,207)
( 41,213)( 42,214)( 43,216)( 44,215)( 45,209)( 46,210)( 47,212)( 48,211)
( 49,193)( 50,194)( 51,196)( 52,195)( 53,201)( 54,202)( 55,204)( 56,203)
( 57,197)( 58,198)( 59,200)( 60,199)( 61,181)( 62,182)( 63,184)( 64,183)
( 65,189)( 66,190)( 67,192)( 68,191)( 69,185)( 70,186)( 71,188)( 72,187)
( 73,169)( 74,170)( 75,172)( 76,171)( 77,177)( 78,178)( 79,180)( 80,179)
( 81,173)( 82,174)( 83,176)( 84,175)( 85,157)( 86,158)( 87,160)( 88,159)
( 89,165)( 90,166)( 91,168)( 92,167)( 93,161)( 94,162)( 95,164)( 96,163)
( 97,145)( 98,146)( 99,148)(100,147)(101,153)(102,154)(103,156)(104,155)
(105,149)(106,150)(107,152)(108,151)(217,313)(218,314)(219,316)(220,315)
(221,321)(222,322)(223,324)(224,323)(225,317)(226,318)(227,320)(228,319)
(229,301)(230,302)(231,304)(232,303)(233,309)(234,310)(235,312)(236,311)
(237,305)(238,306)(239,308)(240,307)(241,289)(242,290)(243,292)(244,291)
(245,297)(246,298)(247,300)(248,299)(249,293)(250,294)(251,296)(252,295)
(253,277)(254,278)(255,280)(256,279)(257,285)(258,286)(259,288)(260,287)
(261,281)(262,282)(263,284)(264,283)(267,268)(269,273)(270,274)(271,276)
(272,275);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239)
(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)(254,255)
(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)(270,271)
(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)(286,287)
(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)(302,303)
(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)(318,319)
(321,324)(322,323);;
s3 := (325,326);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(326)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)
( 43,106)( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)
( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)
( 59, 90)( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)
( 67, 82)( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(109,313)(110,315)
(111,314)(112,316)(113,321)(114,323)(115,322)(116,324)(117,317)(118,319)
(119,318)(120,320)(121,301)(122,303)(123,302)(124,304)(125,309)(126,311)
(127,310)(128,312)(129,305)(130,307)(131,306)(132,308)(133,289)(134,291)
(135,290)(136,292)(137,297)(138,299)(139,298)(140,300)(141,293)(142,295)
(143,294)(144,296)(145,277)(146,279)(147,278)(148,280)(149,285)(150,287)
(151,286)(152,288)(153,281)(154,283)(155,282)(156,284)(157,265)(158,267)
(159,266)(160,268)(161,273)(162,275)(163,274)(164,276)(165,269)(166,271)
(167,270)(168,272)(169,253)(170,255)(171,254)(172,256)(173,261)(174,263)
(175,262)(176,264)(177,257)(178,259)(179,258)(180,260)(181,241)(182,243)
(183,242)(184,244)(185,249)(186,251)(187,250)(188,252)(189,245)(190,247)
(191,246)(192,248)(193,229)(194,231)(195,230)(196,232)(197,237)(198,239)
(199,238)(200,240)(201,233)(202,235)(203,234)(204,236)(205,217)(206,219)
(207,218)(208,220)(209,225)(210,227)(211,226)(212,228)(213,221)(214,223)
(215,222)(216,224);
s1 := Sym(326)!(  1,109)(  2,110)(  3,112)(  4,111)(  5,117)(  6,118)(  7,120)
(  8,119)(  9,113)( 10,114)( 11,116)( 12,115)( 13,137)( 14,138)( 15,140)
( 16,139)( 17,133)( 18,134)( 19,136)( 20,135)( 21,141)( 22,142)( 23,144)
( 24,143)( 25,125)( 26,126)( 27,128)( 28,127)( 29,121)( 30,122)( 31,124)
( 32,123)( 33,129)( 34,130)( 35,132)( 36,131)( 37,205)( 38,206)( 39,208)
( 40,207)( 41,213)( 42,214)( 43,216)( 44,215)( 45,209)( 46,210)( 47,212)
( 48,211)( 49,193)( 50,194)( 51,196)( 52,195)( 53,201)( 54,202)( 55,204)
( 56,203)( 57,197)( 58,198)( 59,200)( 60,199)( 61,181)( 62,182)( 63,184)
( 64,183)( 65,189)( 66,190)( 67,192)( 68,191)( 69,185)( 70,186)( 71,188)
( 72,187)( 73,169)( 74,170)( 75,172)( 76,171)( 77,177)( 78,178)( 79,180)
( 80,179)( 81,173)( 82,174)( 83,176)( 84,175)( 85,157)( 86,158)( 87,160)
( 88,159)( 89,165)( 90,166)( 91,168)( 92,167)( 93,161)( 94,162)( 95,164)
( 96,163)( 97,145)( 98,146)( 99,148)(100,147)(101,153)(102,154)(103,156)
(104,155)(105,149)(106,150)(107,152)(108,151)(217,313)(218,314)(219,316)
(220,315)(221,321)(222,322)(223,324)(224,323)(225,317)(226,318)(227,320)
(228,319)(229,301)(230,302)(231,304)(232,303)(233,309)(234,310)(235,312)
(236,311)(237,305)(238,306)(239,308)(240,307)(241,289)(242,290)(243,292)
(244,291)(245,297)(246,298)(247,300)(248,299)(249,293)(250,294)(251,296)
(252,295)(253,277)(254,278)(255,280)(256,279)(257,285)(258,286)(259,288)
(260,287)(261,281)(262,282)(263,284)(264,283)(267,268)(269,273)(270,274)
(271,276)(272,275);
s2 := Sym(326)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239)(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)
(254,255)(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)
(270,271)(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)
(286,287)(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)
(302,303)(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)
(318,319)(321,324)(322,323);
s3 := Sym(326)!(325,326);
poly := sub<Sym(326)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*s1*s2*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope