Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1296b
if this polytope has a name.
Group : SmallGroup(1296,840)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 54, 324, 108
Order of s₀s₁s₂ : 36
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 : {6,6}*648a
   3-fold quotients : {6,12}*432b
   4-fold quotients : {6,6}*324b
   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, 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)(  4,178)(  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)( 19,181)( 20,183)( 21,182)( 22,187)( 23,189)( 24,188)
( 25,184)( 26,186)( 27,185)( 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)( 46,208)( 47,210)( 48,209)
( 49,214)( 50,216)( 51,215)( 52,211)( 53,213)( 54,212)( 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)
( 73,235)( 74,237)( 75,236)( 76,241)( 77,243)( 78,242)( 79,238)( 80,240)
( 81,239)( 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)(100,262)(101,264)(102,263)(103,268)(104,270)
(105,269)(106,265)(107,267)(108,266)(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)(127,289)(128,291)
(129,290)(130,295)(131,297)(132,296)(133,292)(134,294)(135,293)(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)(154,316)(155,318)(156,317)(157,322)(158,324)(159,323)(160,319)
(161,321)(162,320);;
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, 
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,  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)(  4,178)(  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)( 19,181)( 20,183)( 21,182)( 22,187)( 23,189)
( 24,188)( 25,184)( 26,186)( 27,185)( 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)( 46,208)( 47,210)
( 48,209)( 49,214)( 50,216)( 51,215)( 52,211)( 53,213)( 54,212)( 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)( 73,235)( 74,237)( 75,236)( 76,241)( 77,243)( 78,242)( 79,238)
( 80,240)( 81,239)( 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)(100,262)(101,264)(102,263)(103,268)
(104,270)(105,269)(106,265)(107,267)(108,266)(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)(127,289)
(128,291)(129,290)(130,295)(131,297)(132,296)(133,292)(134,294)(135,293)
(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)(154,316)(155,318)(156,317)(157,322)(158,324)(159,323)
(160,319)(161,321)(162,320);
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, 
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