Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,812)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,6}*648b
   3-fold quotients : {36,6}*432a, {12,6}*432b
   4-fold quotients : {18,6}*324a
   6-fold quotients : {18,6}*216a, {6,6}*216b
   9-fold quotients : {36,2}*144, {12,6}*144a
   12-fold quotients : {6,6}*108
   18-fold quotients : {18,2}*72, {6,6}*72a
   27-fold quotients : {12,2}*48, {4,6}*48a
   36-fold quotients : {9,2}*36
   54-fold quotients : {2,6}*24, {6,2}*24
   81-fold quotients : {4,2}*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 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 25)( 11, 27)( 12, 26)( 13, 22)
( 14, 24)( 15, 23)( 16, 19)( 17, 21)( 18, 20)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 52)( 38, 54)( 39, 53)( 40, 49)( 41, 51)( 42, 50)( 43, 46)
( 44, 48)( 45, 47)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 79)( 65, 81)
( 66, 80)( 67, 76)( 68, 78)( 69, 77)( 70, 73)( 71, 75)( 72, 74)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,106)( 92,108)( 93,107)( 94,103)( 95,105)
( 96,104)( 97,100)( 98,102)( 99,101)(110,111)(112,115)(113,117)(114,116)
(118,133)(119,135)(120,134)(121,130)(122,132)(123,131)(124,127)(125,129)
(126,128)(137,138)(139,142)(140,144)(141,143)(145,160)(146,162)(147,161)
(148,157)(149,159)(150,158)(151,154)(152,156)(153,155)(163,244)(164,246)
(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,268)
(173,270)(174,269)(175,265)(176,267)(177,266)(178,262)(179,264)(180,263)
(181,259)(182,261)(183,260)(184,256)(185,258)(186,257)(187,253)(188,255)
(189,254)(190,271)(191,273)(192,272)(193,277)(194,279)(195,278)(196,274)
(197,276)(198,275)(199,295)(200,297)(201,296)(202,292)(203,294)(204,293)
(205,289)(206,291)(207,290)(208,286)(209,288)(210,287)(211,283)(212,285)
(213,284)(214,280)(215,282)(216,281)(217,298)(218,300)(219,299)(220,304)
(221,306)(222,305)(223,301)(224,303)(225,302)(226,322)(227,324)(228,323)
(229,319)(230,321)(231,320)(232,316)(233,318)(234,317)(235,313)(236,315)
(237,314)(238,310)(239,312)(240,311)(241,307)(242,309)(243,308);;
s1 := (  1,172)(  2,173)(  3,174)(  4,178)(  5,179)(  6,180)(  7,175)(  8,176)
(  9,177)( 10,163)( 11,164)( 12,165)( 13,169)( 14,170)( 15,171)( 16,166)
( 17,167)( 18,168)( 19,187)( 20,188)( 21,189)( 22,184)( 23,185)( 24,186)
( 25,181)( 26,182)( 27,183)( 28,226)( 29,227)( 30,228)( 31,232)( 32,233)
( 33,234)( 34,229)( 35,230)( 36,231)( 37,217)( 38,218)( 39,219)( 40,223)
( 41,224)( 42,225)( 43,220)( 44,221)( 45,222)( 46,241)( 47,242)( 48,243)
( 49,238)( 50,239)( 51,240)( 52,235)( 53,236)( 54,237)( 55,199)( 56,200)
( 57,201)( 58,205)( 59,206)( 60,207)( 61,202)( 62,203)( 63,204)( 64,190)
( 65,191)( 66,192)( 67,196)( 68,197)( 69,198)( 70,193)( 71,194)( 72,195)
( 73,214)( 74,215)( 75,216)( 76,211)( 77,212)( 78,213)( 79,208)( 80,209)
( 81,210)( 82,253)( 83,254)( 84,255)( 85,259)( 86,260)( 87,261)( 88,256)
( 89,257)( 90,258)( 91,244)( 92,245)( 93,246)( 94,250)( 95,251)( 96,252)
( 97,247)( 98,248)( 99,249)(100,268)(101,269)(102,270)(103,265)(104,266)
(105,267)(106,262)(107,263)(108,264)(109,307)(110,308)(111,309)(112,313)
(113,314)(114,315)(115,310)(116,311)(117,312)(118,298)(119,299)(120,300)
(121,304)(122,305)(123,306)(124,301)(125,302)(126,303)(127,322)(128,323)
(129,324)(130,319)(131,320)(132,321)(133,316)(134,317)(135,318)(136,280)
(137,281)(138,282)(139,286)(140,287)(141,288)(142,283)(143,284)(144,285)
(145,271)(146,272)(147,273)(148,277)(149,278)(150,279)(151,274)(152,275)
(153,276)(154,295)(155,296)(156,297)(157,292)(158,293)(159,294)(160,289)
(161,290)(162,291);;
s2 := (  1, 28)(  2, 30)(  3, 29)(  4, 31)(  5, 33)(  6, 32)(  7, 34)(  8, 36)
(  9, 35)( 10, 38)( 11, 37)( 12, 39)( 13, 41)( 14, 40)( 15, 42)( 16, 44)
( 17, 43)( 18, 45)( 19, 48)( 20, 47)( 21, 46)( 22, 51)( 23, 50)( 24, 49)
( 25, 54)( 26, 53)( 27, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 65)( 67, 68)
( 70, 71)( 73, 75)( 76, 78)( 79, 81)( 82,109)( 83,111)( 84,110)( 85,112)
( 86,114)( 87,113)( 88,115)( 89,117)( 90,116)( 91,119)( 92,118)( 93,120)
( 94,122)( 95,121)( 96,123)( 97,125)( 98,124)( 99,126)(100,129)(101,128)
(102,127)(103,132)(104,131)(105,130)(106,135)(107,134)(108,133)(137,138)
(140,141)(143,144)(145,146)(148,149)(151,152)(154,156)(157,159)(160,162)
(163,190)(164,192)(165,191)(166,193)(167,195)(168,194)(169,196)(170,198)
(171,197)(172,200)(173,199)(174,201)(175,203)(176,202)(177,204)(178,206)
(179,205)(180,207)(181,210)(182,209)(183,208)(184,213)(185,212)(186,211)
(187,216)(188,215)(189,214)(218,219)(221,222)(224,225)(226,227)(229,230)
(232,233)(235,237)(238,240)(241,243)(244,271)(245,273)(246,272)(247,274)
(248,276)(249,275)(250,277)(251,279)(252,278)(253,281)(254,280)(255,282)
(256,284)(257,283)(258,285)(259,287)(260,286)(261,288)(262,291)(263,290)
(264,289)(265,294)(266,293)(267,292)(268,297)(269,296)(270,295)(299,300)
(302,303)(305,306)(307,308)(310,311)(313,314)(316,318)(319,321)(322,324);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope