Polytope of Type {18,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,36}*1296a
Also Known As : {18,36|2}. if this polytope has another name.
Group : SmallGroup(1296,798)
Rank : 3
Schlafli Type : {18,36}
Number of vertices, edges, etc : 18, 324, 36
Order of s0s1s2 : 36
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 : {18,18}*648a
3-fold quotients : {6,36}*432a, {18,12}*432a
6-fold quotients : {6,18}*216a, {18,6}*216a
9-fold quotients : {2,36}*144, {18,4}*144a, {6,12}*144a
18-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
27-fold quotients : {2,12}*48, {6,4}*48a
36-fold quotients : {2,9}*36, {9,2}*36
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, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)( 34, 55)
( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)( 42, 69)
( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)( 50, 77)
( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)(111,144)
(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)(119,152)
(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)(127,160)
(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)(135,156)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,223)(191,224)(192,225)(193,220)(194,221)(195,222)(196,217)
(197,218)(198,219)(199,232)(200,233)(201,234)(202,229)(203,230)(204,231)
(205,226)(206,227)(207,228)(208,241)(209,242)(210,243)(211,238)(212,239)
(213,240)(214,235)(215,236)(216,237)(247,250)(248,251)(249,252)(256,259)
(257,260)(258,261)(265,268)(266,269)(267,270)(271,304)(272,305)(273,306)
(274,301)(275,302)(276,303)(277,298)(278,299)(279,300)(280,313)(281,314)
(282,315)(283,310)(284,311)(285,312)(286,307)(287,308)(288,309)(289,322)
(290,323)(291,324)(292,319)(293,320)(294,321)(295,316)(296,317)(297,318);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)( 8, 33)
( 9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 55, 61)( 56, 63)( 57, 62)( 59, 60)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,129)( 92,128)( 93,127)( 94,135)( 95,134)( 96,133)( 97,132)
( 98,131)( 99,130)(100,120)(101,119)(102,118)(103,126)(104,125)(105,124)
(106,123)(107,122)(108,121)(136,142)(137,144)(138,143)(140,141)(145,162)
(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)
(163,271)(164,273)(165,272)(166,277)(167,279)(168,278)(169,274)(170,276)
(171,275)(172,291)(173,290)(174,289)(175,297)(176,296)(177,295)(178,294)
(179,293)(180,292)(181,282)(182,281)(183,280)(184,288)(185,287)(186,286)
(187,285)(188,284)(189,283)(190,244)(191,246)(192,245)(193,250)(194,252)
(195,251)(196,247)(197,249)(198,248)(199,264)(200,263)(201,262)(202,270)
(203,269)(204,268)(205,267)(206,266)(207,265)(208,255)(209,254)(210,253)
(211,261)(212,260)(213,259)(214,258)(215,257)(216,256)(217,304)(218,306)
(219,305)(220,301)(221,303)(222,302)(223,298)(224,300)(225,299)(226,324)
(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)(234,316)
(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)(242,308)
(243,307);;
s2 := ( 1,172)( 2,174)( 3,173)( 4,175)( 5,177)( 6,176)( 7,178)( 8,180)
( 9,179)( 10,163)( 11,165)( 12,164)( 13,166)( 14,168)( 15,167)( 16,169)
( 17,171)( 18,170)( 19,183)( 20,182)( 21,181)( 22,186)( 23,185)( 24,184)
( 25,189)( 26,188)( 27,187)( 28,199)( 29,201)( 30,200)( 31,202)( 32,204)
( 33,203)( 34,205)( 35,207)( 36,206)( 37,190)( 38,192)( 39,191)( 40,193)
( 41,195)( 42,194)( 43,196)( 44,198)( 45,197)( 46,210)( 47,209)( 48,208)
( 49,213)( 50,212)( 51,211)( 52,216)( 53,215)( 54,214)( 55,226)( 56,228)
( 57,227)( 58,229)( 59,231)( 60,230)( 61,232)( 62,234)( 63,233)( 64,217)
( 65,219)( 66,218)( 67,220)( 68,222)( 69,221)( 70,223)( 71,225)( 72,224)
( 73,237)( 74,236)( 75,235)( 76,240)( 77,239)( 78,238)( 79,243)( 80,242)
( 81,241)( 82,253)( 83,255)( 84,254)( 85,256)( 86,258)( 87,257)( 88,259)
( 89,261)( 90,260)( 91,244)( 92,246)( 93,245)( 94,247)( 95,249)( 96,248)
( 97,250)( 98,252)( 99,251)(100,264)(101,263)(102,262)(103,267)(104,266)
(105,265)(106,270)(107,269)(108,268)(109,280)(110,282)(111,281)(112,283)
(113,285)(114,284)(115,286)(116,288)(117,287)(118,271)(119,273)(120,272)
(121,274)(122,276)(123,275)(124,277)(125,279)(126,278)(127,291)(128,290)
(129,289)(130,294)(131,293)(132,292)(133,297)(134,296)(135,295)(136,307)
(137,309)(138,308)(139,310)(140,312)(141,311)(142,313)(143,315)(144,314)
(145,298)(146,300)(147,299)(148,301)(149,303)(150,302)(151,304)(152,306)
(153,305)(154,318)(155,317)(156,316)(157,321)(158,320)(159,319)(160,324)
(161,323)(162,322);;
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*s2*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 61)( 29, 62)( 30, 63)( 31, 58)( 32, 59)( 33, 60)
( 34, 55)( 35, 56)( 36, 57)( 37, 70)( 38, 71)( 39, 72)( 40, 67)( 41, 68)
( 42, 69)( 43, 64)( 44, 65)( 45, 66)( 46, 79)( 47, 80)( 48, 81)( 49, 76)
( 50, 77)( 51, 78)( 52, 73)( 53, 74)( 54, 75)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,142)(110,143)
(111,144)(112,139)(113,140)(114,141)(115,136)(116,137)(117,138)(118,151)
(119,152)(120,153)(121,148)(122,149)(123,150)(124,145)(125,146)(126,147)
(127,160)(128,161)(129,162)(130,157)(131,158)(132,159)(133,154)(134,155)
(135,156)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,223)(191,224)(192,225)(193,220)(194,221)(195,222)
(196,217)(197,218)(198,219)(199,232)(200,233)(201,234)(202,229)(203,230)
(204,231)(205,226)(206,227)(207,228)(208,241)(209,242)(210,243)(211,238)
(212,239)(213,240)(214,235)(215,236)(216,237)(247,250)(248,251)(249,252)
(256,259)(257,260)(258,261)(265,268)(266,269)(267,270)(271,304)(272,305)
(273,306)(274,301)(275,302)(276,303)(277,298)(278,299)(279,300)(280,313)
(281,314)(282,315)(283,310)(284,311)(285,312)(286,307)(287,308)(288,309)
(289,322)(290,323)(291,324)(292,319)(293,320)(294,321)(295,316)(296,317)
(297,318);
s1 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 34)( 5, 36)( 6, 35)( 7, 31)
( 8, 33)( 9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 61)( 56, 63)( 57, 62)( 59, 60)
( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)
( 72, 73)( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)
( 89,114)( 90,113)( 91,129)( 92,128)( 93,127)( 94,135)( 95,134)( 96,133)
( 97,132)( 98,131)( 99,130)(100,120)(101,119)(102,118)(103,126)(104,125)
(105,124)(106,123)(107,122)(108,121)(136,142)(137,144)(138,143)(140,141)
(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)
(153,154)(163,271)(164,273)(165,272)(166,277)(167,279)(168,278)(169,274)
(170,276)(171,275)(172,291)(173,290)(174,289)(175,297)(176,296)(177,295)
(178,294)(179,293)(180,292)(181,282)(182,281)(183,280)(184,288)(185,287)
(186,286)(187,285)(188,284)(189,283)(190,244)(191,246)(192,245)(193,250)
(194,252)(195,251)(196,247)(197,249)(198,248)(199,264)(200,263)(201,262)
(202,270)(203,269)(204,268)(205,267)(206,266)(207,265)(208,255)(209,254)
(210,253)(211,261)(212,260)(213,259)(214,258)(215,257)(216,256)(217,304)
(218,306)(219,305)(220,301)(221,303)(222,302)(223,298)(224,300)(225,299)
(226,324)(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)
(234,316)(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)
(242,308)(243,307);
s2 := Sym(324)!( 1,172)( 2,174)( 3,173)( 4,175)( 5,177)( 6,176)( 7,178)
( 8,180)( 9,179)( 10,163)( 11,165)( 12,164)( 13,166)( 14,168)( 15,167)
( 16,169)( 17,171)( 18,170)( 19,183)( 20,182)( 21,181)( 22,186)( 23,185)
( 24,184)( 25,189)( 26,188)( 27,187)( 28,199)( 29,201)( 30,200)( 31,202)
( 32,204)( 33,203)( 34,205)( 35,207)( 36,206)( 37,190)( 38,192)( 39,191)
( 40,193)( 41,195)( 42,194)( 43,196)( 44,198)( 45,197)( 46,210)( 47,209)
( 48,208)( 49,213)( 50,212)( 51,211)( 52,216)( 53,215)( 54,214)( 55,226)
( 56,228)( 57,227)( 58,229)( 59,231)( 60,230)( 61,232)( 62,234)( 63,233)
( 64,217)( 65,219)( 66,218)( 67,220)( 68,222)( 69,221)( 70,223)( 71,225)
( 72,224)( 73,237)( 74,236)( 75,235)( 76,240)( 77,239)( 78,238)( 79,243)
( 80,242)( 81,241)( 82,253)( 83,255)( 84,254)( 85,256)( 86,258)( 87,257)
( 88,259)( 89,261)( 90,260)( 91,244)( 92,246)( 93,245)( 94,247)( 95,249)
( 96,248)( 97,250)( 98,252)( 99,251)(100,264)(101,263)(102,262)(103,267)
(104,266)(105,265)(106,270)(107,269)(108,268)(109,280)(110,282)(111,281)
(112,283)(113,285)(114,284)(115,286)(116,288)(117,287)(118,271)(119,273)
(120,272)(121,274)(122,276)(123,275)(124,277)(125,279)(126,278)(127,291)
(128,290)(129,289)(130,294)(131,293)(132,292)(133,297)(134,296)(135,295)
(136,307)(137,309)(138,308)(139,310)(140,312)(141,311)(142,313)(143,315)
(144,314)(145,298)(146,300)(147,299)(148,301)(149,303)(150,302)(151,304)
(152,306)(153,305)(154,318)(155,317)(156,316)(157,321)(158,320)(159,319)
(160,324)(161,323)(162,322);
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*s2*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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