Polytope of Type {6,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1296g
if this polytope has a name.
Group : SmallGroup(1296,868)
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}*648g
3-fold quotients : {6,12}*432b
4-fold quotients : {6,18}*324c
6-fold quotients : {6,6}*216b
9-fold quotients : {6,12}*144a
12-fold quotients : {6,6}*108
18-fold quotients : {6,6}*72a
27-fold quotients : {2,12}*48, {6,4}*48a
54-fold quotients : {2,6}*24, {6,2}*24
81-fold quotients : {2,4}*16
108-fold quotients : {2,3}*12, {3,2}*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, 53)( 11, 52)( 12, 54)( 13, 47)( 14, 46)( 15, 48)( 16, 50)
( 17, 49)( 18, 51)( 19, 41)( 20, 40)( 21, 42)( 22, 44)( 23, 43)( 24, 45)
( 25, 38)( 26, 37)( 27, 39)( 56, 57)( 59, 60)( 62, 63)( 64, 80)( 65, 79)
( 66, 81)( 67, 74)( 68, 73)( 69, 75)( 70, 77)( 71, 76)( 72, 78)( 82,109)
( 83,111)( 84,110)( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)
( 91,134)( 92,133)( 93,135)( 94,128)( 95,127)( 96,129)( 97,131)( 98,130)
( 99,132)(100,122)(101,121)(102,123)(103,125)(104,124)(105,126)(106,119)
(107,118)(108,120)(137,138)(140,141)(143,144)(145,161)(146,160)(147,162)
(148,155)(149,154)(150,156)(151,158)(152,157)(153,159)(163,271)(164,273)
(165,272)(166,274)(167,276)(168,275)(169,277)(170,279)(171,278)(172,296)
(173,295)(174,297)(175,290)(176,289)(177,291)(178,293)(179,292)(180,294)
(181,284)(182,283)(183,285)(184,287)(185,286)(186,288)(187,281)(188,280)
(189,282)(190,244)(191,246)(192,245)(193,247)(194,249)(195,248)(196,250)
(197,252)(198,251)(199,269)(200,268)(201,270)(202,263)(203,262)(204,264)
(205,266)(206,265)(207,267)(208,257)(209,256)(210,258)(211,260)(212,259)
(213,261)(214,254)(215,253)(216,255)(217,298)(218,300)(219,299)(220,301)
(221,303)(222,302)(223,304)(224,306)(225,305)(226,323)(227,322)(228,324)
(229,317)(230,316)(231,318)(232,320)(233,319)(234,321)(235,311)(236,310)
(237,312)(238,314)(239,313)(240,315)(241,308)(242,307)(243,309);;
s2 := ( 1,172)( 2,174)( 3,173)( 4,178)( 5,180)( 6,179)( 7,175)( 8,177)
( 9,176)( 10,163)( 11,165)( 12,164)( 13,169)( 14,171)( 15,170)( 16,166)
( 17,168)( 18,167)( 19,182)( 20,181)( 21,183)( 22,188)( 23,187)( 24,189)
( 25,185)( 26,184)( 27,186)( 28,199)( 29,201)( 30,200)( 31,205)( 32,207)
( 33,206)( 34,202)( 35,204)( 36,203)( 37,190)( 38,192)( 39,191)( 40,196)
( 41,198)( 42,197)( 43,193)( 44,195)( 45,194)( 46,209)( 47,208)( 48,210)
( 49,215)( 50,214)( 51,216)( 52,212)( 53,211)( 54,213)( 55,226)( 56,228)
( 57,227)( 58,232)( 59,234)( 60,233)( 61,229)( 62,231)( 63,230)( 64,217)
( 65,219)( 66,218)( 67,223)( 68,225)( 69,224)( 70,220)( 71,222)( 72,221)
( 73,236)( 74,235)( 75,237)( 76,242)( 77,241)( 78,243)( 79,239)( 80,238)
( 81,240)( 82,253)( 83,255)( 84,254)( 85,259)( 86,261)( 87,260)( 88,256)
( 89,258)( 90,257)( 91,244)( 92,246)( 93,245)( 94,250)( 95,252)( 96,251)
( 97,247)( 98,249)( 99,248)(100,263)(101,262)(102,264)(103,269)(104,268)
(105,270)(106,266)(107,265)(108,267)(109,280)(110,282)(111,281)(112,286)
(113,288)(114,287)(115,283)(116,285)(117,284)(118,271)(119,273)(120,272)
(121,277)(122,279)(123,278)(124,274)(125,276)(126,275)(127,290)(128,289)
(129,291)(130,296)(131,295)(132,297)(133,293)(134,292)(135,294)(136,307)
(137,309)(138,308)(139,313)(140,315)(141,314)(142,310)(143,312)(144,311)
(145,298)(146,300)(147,299)(148,304)(149,306)(150,305)(151,301)(152,303)
(153,302)(154,317)(155,316)(156,318)(157,323)(158,322)(159,324)(160,320)
(161,319)(162,321);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*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, 53)( 11, 52)( 12, 54)( 13, 47)( 14, 46)( 15, 48)
( 16, 50)( 17, 49)( 18, 51)( 19, 41)( 20, 40)( 21, 42)( 22, 44)( 23, 43)
( 24, 45)( 25, 38)( 26, 37)( 27, 39)( 56, 57)( 59, 60)( 62, 63)( 64, 80)
( 65, 79)( 66, 81)( 67, 74)( 68, 73)( 69, 75)( 70, 77)( 71, 76)( 72, 78)
( 82,109)( 83,111)( 84,110)( 85,112)( 86,114)( 87,113)( 88,115)( 89,117)
( 90,116)( 91,134)( 92,133)( 93,135)( 94,128)( 95,127)( 96,129)( 97,131)
( 98,130)( 99,132)(100,122)(101,121)(102,123)(103,125)(104,124)(105,126)
(106,119)(107,118)(108,120)(137,138)(140,141)(143,144)(145,161)(146,160)
(147,162)(148,155)(149,154)(150,156)(151,158)(152,157)(153,159)(163,271)
(164,273)(165,272)(166,274)(167,276)(168,275)(169,277)(170,279)(171,278)
(172,296)(173,295)(174,297)(175,290)(176,289)(177,291)(178,293)(179,292)
(180,294)(181,284)(182,283)(183,285)(184,287)(185,286)(186,288)(187,281)
(188,280)(189,282)(190,244)(191,246)(192,245)(193,247)(194,249)(195,248)
(196,250)(197,252)(198,251)(199,269)(200,268)(201,270)(202,263)(203,262)
(204,264)(205,266)(206,265)(207,267)(208,257)(209,256)(210,258)(211,260)
(212,259)(213,261)(214,254)(215,253)(216,255)(217,298)(218,300)(219,299)
(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)(226,323)(227,322)
(228,324)(229,317)(230,316)(231,318)(232,320)(233,319)(234,321)(235,311)
(236,310)(237,312)(238,314)(239,313)(240,315)(241,308)(242,307)(243,309);
s2 := Sym(324)!( 1,172)( 2,174)( 3,173)( 4,178)( 5,180)( 6,179)( 7,175)
( 8,177)( 9,176)( 10,163)( 11,165)( 12,164)( 13,169)( 14,171)( 15,170)
( 16,166)( 17,168)( 18,167)( 19,182)( 20,181)( 21,183)( 22,188)( 23,187)
( 24,189)( 25,185)( 26,184)( 27,186)( 28,199)( 29,201)( 30,200)( 31,205)
( 32,207)( 33,206)( 34,202)( 35,204)( 36,203)( 37,190)( 38,192)( 39,191)
( 40,196)( 41,198)( 42,197)( 43,193)( 44,195)( 45,194)( 46,209)( 47,208)
( 48,210)( 49,215)( 50,214)( 51,216)( 52,212)( 53,211)( 54,213)( 55,226)
( 56,228)( 57,227)( 58,232)( 59,234)( 60,233)( 61,229)( 62,231)( 63,230)
( 64,217)( 65,219)( 66,218)( 67,223)( 68,225)( 69,224)( 70,220)( 71,222)
( 72,221)( 73,236)( 74,235)( 75,237)( 76,242)( 77,241)( 78,243)( 79,239)
( 80,238)( 81,240)( 82,253)( 83,255)( 84,254)( 85,259)( 86,261)( 87,260)
( 88,256)( 89,258)( 90,257)( 91,244)( 92,246)( 93,245)( 94,250)( 95,252)
( 96,251)( 97,247)( 98,249)( 99,248)(100,263)(101,262)(102,264)(103,269)
(104,268)(105,270)(106,266)(107,265)(108,267)(109,280)(110,282)(111,281)
(112,286)(113,288)(114,287)(115,283)(116,285)(117,284)(118,271)(119,273)
(120,272)(121,277)(122,279)(123,278)(124,274)(125,276)(126,275)(127,290)
(128,289)(129,291)(130,296)(131,295)(132,297)(133,293)(134,292)(135,294)
(136,307)(137,309)(138,308)(139,313)(140,315)(141,314)(142,310)(143,312)
(144,311)(145,298)(146,300)(147,299)(148,304)(149,306)(150,305)(151,301)
(152,303)(153,302)(154,317)(155,316)(156,318)(157,323)(158,322)(159,324)
(160,320)(161,319)(162,321);
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope