Polytope of Type {18,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,36}*1296b
if this polytope has a 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 : 18
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}*648b
   3-fold quotients : {6,36}*432b
   4-fold quotients : {18,9}*324
   6-fold quotients : {6,18}*216b
   9-fold quotients : {2,36}*144, {6,12}*144b
   12-fold quotients : {6,9}*108
   18-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {2,12}*48
   36-fold quotients : {2,9}*36, {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,  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)(  4,178)(  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)( 19,183)( 20,182)( 21,181)( 22,189)( 23,188)( 24,187)
( 25,186)( 26,185)( 27,184)( 28,232)( 29,234)( 30,233)( 31,229)( 32,231)
( 33,230)( 34,226)( 35,228)( 36,227)( 37,223)( 38,225)( 39,224)( 40,220)
( 41,222)( 42,221)( 43,217)( 44,219)( 45,218)( 46,243)( 47,242)( 48,241)
( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,205)( 56,207)
( 57,206)( 58,202)( 59,204)( 60,203)( 61,199)( 62,201)( 63,200)( 64,196)
( 65,198)( 66,197)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)( 72,191)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)( 80,209)
( 81,208)( 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)(100,264)(101,263)(102,262)(103,270)(104,269)
(105,268)(106,267)(107,266)(108,265)(109,313)(110,315)(111,314)(112,310)
(113,312)(114,311)(115,307)(116,309)(117,308)(118,304)(119,306)(120,305)
(121,301)(122,303)(123,302)(124,298)(125,300)(126,299)(127,324)(128,323)
(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)(136,286)
(137,288)(138,287)(139,283)(140,285)(141,284)(142,280)(143,282)(144,281)
(145,277)(146,279)(147,278)(148,274)(149,276)(150,275)(151,271)(152,273)
(153,272)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)(160,291)
(161,290)(162,289);;
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, s2*s0*s1*s0*s1*s2*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*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)(  4,178)(  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)( 19,183)( 20,182)( 21,181)( 22,189)( 23,188)
( 24,187)( 25,186)( 26,185)( 27,184)( 28,232)( 29,234)( 30,233)( 31,229)
( 32,231)( 33,230)( 34,226)( 35,228)( 36,227)( 37,223)( 38,225)( 39,224)
( 40,220)( 41,222)( 42,221)( 43,217)( 44,219)( 45,218)( 46,243)( 47,242)
( 48,241)( 49,240)( 50,239)( 51,238)( 52,237)( 53,236)( 54,235)( 55,205)
( 56,207)( 57,206)( 58,202)( 59,204)( 60,203)( 61,199)( 62,201)( 63,200)
( 64,196)( 65,198)( 66,197)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)
( 72,191)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,210)
( 80,209)( 81,208)( 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)(100,264)(101,263)(102,262)(103,270)
(104,269)(105,268)(106,267)(107,266)(108,265)(109,313)(110,315)(111,314)
(112,310)(113,312)(114,311)(115,307)(116,309)(117,308)(118,304)(119,306)
(120,305)(121,301)(122,303)(123,302)(124,298)(125,300)(126,299)(127,324)
(128,323)(129,322)(130,321)(131,320)(132,319)(133,318)(134,317)(135,316)
(136,286)(137,288)(138,287)(139,283)(140,285)(141,284)(142,280)(143,282)
(144,281)(145,277)(146,279)(147,278)(148,274)(149,276)(150,275)(151,271)
(152,273)(153,272)(154,297)(155,296)(156,295)(157,294)(158,293)(159,292)
(160,291)(161,290)(162,289);
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, s2*s0*s1*s0*s1*s2*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*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