Polytope of Type {2,4,81}

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

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

Permutation Representation (Magma) :

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

to this polytope