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# Polytope of Type {6,36}

Atlas Canonical Name : {6,36}*1296d
if this polytope has a name.
Group : SmallGroup(1296,840)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,18}*648c
3-fold quotients : {6,12}*432a
4-fold quotients : {6,9}*324c
6-fold quotients : {6,6}*216a
9-fold quotients : {6,12}*144b
12-fold quotients : {6,3}*108
18-fold quotients : {6,6}*72b
27-fold quotients : {2,12}*48
36-fold quotients : {6,3}*36
54-fold quotients : {2,6}*24
81-fold quotients : {2,4}*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 := (  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)
( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)
( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)
( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)
( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)
( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)
(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)
(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)
(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)
(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)
(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)
(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)
(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)
(213,241)(214,240)(215,238)(216,239)(247,251)(248,252)(249,250)(256,260)
(257,261)(258,259)(265,269)(266,270)(267,268)(271,298)(272,299)(273,300)
(274,305)(275,306)(276,304)(277,303)(278,301)(279,302)(280,307)(281,308)
(282,309)(283,314)(284,315)(285,313)(286,312)(287,310)(288,311)(289,316)
(290,317)(291,318)(292,323)(293,324)(294,322)(295,321)(296,319)(297,320);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 52)( 11, 54)( 12, 53)( 13, 46)( 14, 48)( 15, 47)( 16, 49)
( 17, 51)( 18, 50)( 19, 40)( 20, 42)( 21, 41)( 22, 43)( 23, 45)( 24, 44)
( 25, 37)( 26, 39)( 27, 38)( 56, 57)( 59, 60)( 62, 63)( 64, 79)( 65, 81)
( 66, 80)( 67, 73)( 68, 75)( 69, 74)( 70, 76)( 71, 78)( 72, 77)( 82,109)
( 83,111)( 84,110)( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)
( 91,133)( 92,135)( 93,134)( 94,127)( 95,129)( 96,128)( 97,130)( 98,132)
( 99,131)(100,121)(101,123)(102,122)(103,124)(104,126)(105,125)(106,118)
(107,120)(108,119)(137,138)(140,141)(143,144)(145,160)(146,162)(147,161)
(148,154)(149,156)(150,155)(151,157)(152,159)(153,158)(163,271)(164,273)
(165,272)(166,274)(167,276)(168,275)(169,277)(170,279)(171,278)(172,295)
(173,297)(174,296)(175,289)(176,291)(177,290)(178,292)(179,294)(180,293)
(181,283)(182,285)(183,284)(184,286)(185,288)(186,287)(187,280)(188,282)
(189,281)(190,244)(191,246)(192,245)(193,247)(194,249)(195,248)(196,250)
(197,252)(198,251)(199,268)(200,270)(201,269)(202,262)(203,264)(204,263)
(205,265)(206,267)(207,266)(208,256)(209,258)(210,257)(211,259)(212,261)
(213,260)(214,253)(215,255)(216,254)(217,298)(218,300)(219,299)(220,301)
(221,303)(222,302)(223,304)(224,306)(225,305)(226,322)(227,324)(228,323)
(229,316)(230,318)(231,317)(232,319)(233,321)(234,320)(235,310)(236,312)
(237,311)(238,313)(239,315)(240,314)(241,307)(242,309)(243,308);;
s2 := (  1,172)(  2,174)(  3,173)(  4,177)(  5,176)(  6,175)(  7,179)(  8,178)
(  9,180)( 10,163)( 11,165)( 12,164)( 13,168)( 14,167)( 15,166)( 16,170)
( 17,169)( 18,171)( 19,181)( 20,183)( 21,182)( 22,186)( 23,185)( 24,184)
( 25,188)( 26,187)( 27,189)( 28,226)( 29,228)( 30,227)( 31,231)( 32,230)
( 33,229)( 34,233)( 35,232)( 36,234)( 37,217)( 38,219)( 39,218)( 40,222)
( 41,221)( 42,220)( 43,224)( 44,223)( 45,225)( 46,235)( 47,237)( 48,236)
( 49,240)( 50,239)( 51,238)( 52,242)( 53,241)( 54,243)( 55,199)( 56,201)
( 57,200)( 58,204)( 59,203)( 60,202)( 61,206)( 62,205)( 63,207)( 64,190)
( 65,192)( 66,191)( 67,195)( 68,194)( 69,193)( 70,197)( 71,196)( 72,198)
( 73,208)( 74,210)( 75,209)( 76,213)( 77,212)( 78,211)( 79,215)( 80,214)
( 81,216)( 82,253)( 83,255)( 84,254)( 85,258)( 86,257)( 87,256)( 88,260)
( 89,259)( 90,261)( 91,244)( 92,246)( 93,245)( 94,249)( 95,248)( 96,247)
( 97,251)( 98,250)( 99,252)(100,262)(101,264)(102,263)(103,267)(104,266)
(105,265)(106,269)(107,268)(108,270)(109,307)(110,309)(111,308)(112,312)
(113,311)(114,310)(115,314)(116,313)(117,315)(118,298)(119,300)(120,299)
(121,303)(122,302)(123,301)(124,305)(125,304)(126,306)(127,316)(128,318)
(129,317)(130,321)(131,320)(132,319)(133,323)(134,322)(135,324)(136,280)
(137,282)(138,281)(139,285)(140,284)(141,283)(142,287)(143,286)(144,288)
(145,271)(146,273)(147,272)(148,276)(149,275)(150,274)(151,278)(152,277)
(153,279)(154,289)(155,291)(156,290)(157,294)(158,293)(159,292)(160,296)
(161,295)(162,297);;
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*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

References : None.
to this polytope