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Polytope of Type {4,162}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,162}*1296a
Also Known As : {4,162|2}. if this polytope has another name.
Group : SmallGroup(1296,67)
Rank : 3
Schlafli Type : {4,162}
Number of vertices, edges, etc : 4, 324, 162
Order of s0s1s2 : 324
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   2-fold quotients : {2,162}*648
   3-fold quotients : {4,54}*432a
   4-fold quotients : {2,81}*324
   6-fold quotients : {2,54}*216
   9-fold quotients : {4,18}*144a
   12-fold quotients : {2,27}*108
   18-fold quotients : {2,18}*72
   27-fold quotients : {4,6}*48a
   36-fold quotients : {2,9}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (163,244)(164,245)(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)
(171,252)(172,253)(173,254)(174,255)(175,256)(176,257)(177,258)(178,259)
(179,260)(180,261)(181,262)(182,263)(183,264)(184,265)(185,266)(186,267)
(187,268)(188,269)(189,270)(190,271)(191,272)(192,273)(193,274)(194,275)
(195,276)(196,277)(197,278)(198,279)(199,280)(200,281)(201,282)(202,283)
(203,284)(204,285)(205,286)(206,287)(207,288)(208,289)(209,290)(210,291)
(211,292)(212,293)(213,294)(214,295)(215,296)(216,297)(217,298)(218,299)
(219,300)(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)(226,307)
(227,308)(228,309)(229,310)(230,311)(231,312)(232,313)(233,314)(234,315)
(235,316)(236,317)(237,318)(238,319)(239,320)(240,321)(241,322)(242,323)
(243,324);;
s1 := (  1,163)(  2,165)(  3,164)(  4,171)(  5,170)(  6,169)(  7,168)(  8,167)
(  9,166)( 10,189)( 11,188)( 12,187)( 13,186)( 14,185)( 15,184)( 16,183)
( 17,182)( 18,181)( 19,180)( 20,179)( 21,178)( 22,177)( 23,176)( 24,175)
( 25,174)( 26,173)( 27,172)( 28,243)( 29,242)( 30,241)( 31,240)( 32,239)
( 33,238)( 34,237)( 35,236)( 36,235)( 37,234)( 38,233)( 39,232)( 40,231)
( 41,230)( 42,229)( 43,228)( 44,227)( 45,226)( 46,225)( 47,224)( 48,223)
( 49,222)( 50,221)( 51,220)( 52,219)( 53,218)( 54,217)( 55,216)( 56,215)
( 57,214)( 58,213)( 59,212)( 60,211)( 61,210)( 62,209)( 63,208)( 64,207)
( 65,206)( 66,205)( 67,204)( 68,203)( 69,202)( 70,201)( 71,200)( 72,199)
( 73,198)( 74,197)( 75,196)( 76,195)( 77,194)( 78,193)( 79,192)( 80,191)
( 81,190)( 82,244)( 83,246)( 84,245)( 85,252)( 86,251)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,270)( 92,269)( 93,268)( 94,267)( 95,266)( 96,265)
( 97,264)( 98,263)( 99,262)(100,261)(101,260)(102,259)(103,258)(104,257)
(105,256)(106,255)(107,254)(108,253)(109,324)(110,323)(111,322)(112,321)
(113,320)(114,319)(115,318)(116,317)(117,316)(118,315)(119,314)(120,313)
(121,312)(122,311)(123,310)(124,309)(125,308)(126,307)(127,306)(128,305)
(129,304)(130,303)(131,302)(132,301)(133,300)(134,299)(135,298)(136,297)
(137,296)(138,295)(139,294)(140,293)(141,292)(142,291)(143,290)(144,289)
(145,288)(146,287)(147,286)(148,285)(149,284)(150,283)(151,282)(152,281)
(153,280)(154,279)(155,278)(156,277)(157,276)(158,275)(159,274)(160,273)
(161,272)(162,271);;
s2 := (  1, 28)(  2, 30)(  3, 29)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)
(  9, 31)( 10, 54)( 11, 53)( 12, 52)( 13, 51)( 14, 50)( 15, 49)( 16, 48)
( 17, 47)( 18, 46)( 19, 45)( 20, 44)( 21, 43)( 22, 42)( 23, 41)( 24, 40)
( 25, 39)( 26, 38)( 27, 37)( 55, 81)( 56, 80)( 57, 79)( 58, 78)( 59, 77)
( 60, 76)( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)
( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)
( 90,112)( 91,135)( 92,134)( 93,133)( 94,132)( 95,131)( 96,130)( 97,129)
( 98,128)( 99,127)(100,126)(101,125)(102,124)(103,123)(104,122)(105,121)
(106,120)(107,119)(108,118)(136,162)(137,161)(138,160)(139,159)(140,158)
(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)(148,150)
(163,190)(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)(170,194)
(171,193)(172,216)(173,215)(174,214)(175,213)(176,212)(177,211)(178,210)
(179,209)(180,208)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)
(187,201)(188,200)(189,199)(217,243)(218,242)(219,241)(220,240)(221,239)
(222,238)(223,237)(224,236)(225,235)(226,234)(227,233)(228,232)(229,231)
(244,271)(245,273)(246,272)(247,279)(248,278)(249,277)(250,276)(251,275)
(252,274)(253,297)(254,296)(255,295)(256,294)(257,293)(258,292)(259,291)
(260,290)(261,289)(262,288)(263,287)(264,286)(265,285)(266,284)(267,283)
(268,282)(269,281)(270,280)(298,324)(299,323)(300,322)(301,321)(302,320)
(303,319)(304,318)(305,317)(306,316)(307,315)(308,314)(309,313)(310,312);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, 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*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(324)!(163,244)(164,245)(165,246)(166,247)(167,248)(168,249)(169,250)
(170,251)(171,252)(172,253)(173,254)(174,255)(175,256)(176,257)(177,258)
(178,259)(179,260)(180,261)(181,262)(182,263)(183,264)(184,265)(185,266)
(186,267)(187,268)(188,269)(189,270)(190,271)(191,272)(192,273)(193,274)
(194,275)(195,276)(196,277)(197,278)(198,279)(199,280)(200,281)(201,282)
(202,283)(203,284)(204,285)(205,286)(206,287)(207,288)(208,289)(209,290)
(210,291)(211,292)(212,293)(213,294)(214,295)(215,296)(216,297)(217,298)
(218,299)(219,300)(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)
(226,307)(227,308)(228,309)(229,310)(230,311)(231,312)(232,313)(233,314)
(234,315)(235,316)(236,317)(237,318)(238,319)(239,320)(240,321)(241,322)
(242,323)(243,324);
s1 := Sym(324)!(  1,163)(  2,165)(  3,164)(  4,171)(  5,170)(  6,169)(  7,168)
(  8,167)(  9,166)( 10,189)( 11,188)( 12,187)( 13,186)( 14,185)( 15,184)
( 16,183)( 17,182)( 18,181)( 19,180)( 20,179)( 21,178)( 22,177)( 23,176)
( 24,175)( 25,174)( 26,173)( 27,172)( 28,243)( 29,242)( 30,241)( 31,240)
( 32,239)( 33,238)( 34,237)( 35,236)( 36,235)( 37,234)( 38,233)( 39,232)
( 40,231)( 41,230)( 42,229)( 43,228)( 44,227)( 45,226)( 46,225)( 47,224)
( 48,223)( 49,222)( 50,221)( 51,220)( 52,219)( 53,218)( 54,217)( 55,216)
( 56,215)( 57,214)( 58,213)( 59,212)( 60,211)( 61,210)( 62,209)( 63,208)
( 64,207)( 65,206)( 66,205)( 67,204)( 68,203)( 69,202)( 70,201)( 71,200)
( 72,199)( 73,198)( 74,197)( 75,196)( 76,195)( 77,194)( 78,193)( 79,192)
( 80,191)( 81,190)( 82,244)( 83,246)( 84,245)( 85,252)( 86,251)( 87,250)
( 88,249)( 89,248)( 90,247)( 91,270)( 92,269)( 93,268)( 94,267)( 95,266)
( 96,265)( 97,264)( 98,263)( 99,262)(100,261)(101,260)(102,259)(103,258)
(104,257)(105,256)(106,255)(107,254)(108,253)(109,324)(110,323)(111,322)
(112,321)(113,320)(114,319)(115,318)(116,317)(117,316)(118,315)(119,314)
(120,313)(121,312)(122,311)(123,310)(124,309)(125,308)(126,307)(127,306)
(128,305)(129,304)(130,303)(131,302)(132,301)(133,300)(134,299)(135,298)
(136,297)(137,296)(138,295)(139,294)(140,293)(141,292)(142,291)(143,290)
(144,289)(145,288)(146,287)(147,286)(148,285)(149,284)(150,283)(151,282)
(152,281)(153,280)(154,279)(155,278)(156,277)(157,276)(158,275)(159,274)
(160,273)(161,272)(162,271);
s2 := Sym(324)!(  1, 28)(  2, 30)(  3, 29)(  4, 36)(  5, 35)(  6, 34)(  7, 33)
(  8, 32)(  9, 31)( 10, 54)( 11, 53)( 12, 52)( 13, 51)( 14, 50)( 15, 49)
( 16, 48)( 17, 47)( 18, 46)( 19, 45)( 20, 44)( 21, 43)( 22, 42)( 23, 41)
( 24, 40)( 25, 39)( 26, 38)( 27, 37)( 55, 81)( 56, 80)( 57, 79)( 58, 78)
( 59, 77)( 60, 76)( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)
( 67, 69)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)
( 89,113)( 90,112)( 91,135)( 92,134)( 93,133)( 94,132)( 95,131)( 96,130)
( 97,129)( 98,128)( 99,127)(100,126)(101,125)(102,124)(103,123)(104,122)
(105,121)(106,120)(107,119)(108,118)(136,162)(137,161)(138,160)(139,159)
(140,158)(141,157)(142,156)(143,155)(144,154)(145,153)(146,152)(147,151)
(148,150)(163,190)(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)
(170,194)(171,193)(172,216)(173,215)(174,214)(175,213)(176,212)(177,211)
(178,210)(179,209)(180,208)(181,207)(182,206)(183,205)(184,204)(185,203)
(186,202)(187,201)(188,200)(189,199)(217,243)(218,242)(219,241)(220,240)
(221,239)(222,238)(223,237)(224,236)(225,235)(226,234)(227,233)(228,232)
(229,231)(244,271)(245,273)(246,272)(247,279)(248,278)(249,277)(250,276)
(251,275)(252,274)(253,297)(254,296)(255,295)(256,294)(257,293)(258,292)
(259,291)(260,290)(261,289)(262,288)(263,287)(264,286)(265,285)(266,284)
(267,283)(268,282)(269,281)(270,280)(298,324)(299,323)(300,322)(301,321)
(302,320)(303,319)(304,318)(305,317)(306,316)(307,315)(308,314)(309,313)
(310,312);
poly := sub<Sym(324)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, 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*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 >; 
 
References : None.
to this polytope