Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,868)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
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 : {6,6}*648d
   3-fold quotients : {12,6}*432a
   4-fold quotients : {3,6}*324
   6-fold quotients : {6,6}*216c
   9-fold quotients : {12,6}*144b
   12-fold quotients : {3,6}*108
   18-fold quotients : {6,6}*72c
   27-fold quotients : {12,2}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {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,  6)(  7,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 22)( 14, 24)
( 15, 23)( 16, 27)( 17, 26)( 18, 25)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 74)( 38, 73)( 39, 75)
( 40, 76)( 41, 78)( 42, 77)( 43, 81)( 44, 80)( 45, 79)( 46, 65)( 47, 64)
( 48, 66)( 49, 67)( 50, 69)( 51, 68)( 52, 72)( 53, 71)( 54, 70)( 83, 84)
( 85, 87)( 88, 89)( 91,101)( 92,100)( 93,102)( 94,103)( 95,105)( 96,104)
( 97,108)( 98,107)( 99,106)(109,136)(110,138)(111,137)(112,141)(113,140)
(114,139)(115,143)(116,142)(117,144)(118,155)(119,154)(120,156)(121,157)
(122,159)(123,158)(124,162)(125,161)(126,160)(127,146)(128,145)(129,147)
(130,148)(131,150)(132,149)(133,153)(134,152)(135,151)(163,244)(164,246)
(165,245)(166,249)(167,248)(168,247)(169,251)(170,250)(171,252)(172,263)
(173,262)(174,264)(175,265)(176,267)(177,266)(178,270)(179,269)(180,268)
(181,254)(182,253)(183,255)(184,256)(185,258)(186,257)(187,261)(188,260)
(189,259)(190,298)(191,300)(192,299)(193,303)(194,302)(195,301)(196,305)
(197,304)(198,306)(199,317)(200,316)(201,318)(202,319)(203,321)(204,320)
(205,324)(206,323)(207,322)(208,308)(209,307)(210,309)(211,310)(212,312)
(213,311)(214,315)(215,314)(216,313)(217,271)(218,273)(219,272)(220,276)
(221,275)(222,274)(223,278)(224,277)(225,279)(226,290)(227,289)(228,291)
(229,292)(230,294)(231,293)(232,297)(233,296)(234,295)(235,281)(236,280)
(237,282)(238,283)(239,285)(240,284)(241,288)(242,287)(243,286);;
s1 := (  1,229)(  2,231)(  3,230)(  4,233)(  5,232)(  6,234)(  7,228)(  8,227)
(  9,226)( 10,225)( 11,224)( 12,223)( 13,217)( 14,219)( 15,218)( 16,221)
( 17,220)( 18,222)( 19,237)( 20,236)( 21,235)( 22,238)( 23,240)( 24,239)
( 25,242)( 26,241)( 27,243)( 28,202)( 29,204)( 30,203)( 31,206)( 32,205)
( 33,207)( 34,201)( 35,200)( 36,199)( 37,198)( 38,197)( 39,196)( 40,190)
( 41,192)( 42,191)( 43,194)( 44,193)( 45,195)( 46,210)( 47,209)( 48,208)
( 49,211)( 50,213)( 51,212)( 52,215)( 53,214)( 54,216)( 55,175)( 56,177)
( 57,176)( 58,179)( 59,178)( 60,180)( 61,174)( 62,173)( 63,172)( 64,171)
( 65,170)( 66,169)( 67,163)( 68,165)( 69,164)( 70,167)( 71,166)( 72,168)
( 73,183)( 74,182)( 75,181)( 76,184)( 77,186)( 78,185)( 79,188)( 80,187)
( 81,189)( 82,310)( 83,312)( 84,311)( 85,314)( 86,313)( 87,315)( 88,309)
( 89,308)( 90,307)( 91,306)( 92,305)( 93,304)( 94,298)( 95,300)( 96,299)
( 97,302)( 98,301)( 99,303)(100,318)(101,317)(102,316)(103,319)(104,321)
(105,320)(106,323)(107,322)(108,324)(109,283)(110,285)(111,284)(112,287)
(113,286)(114,288)(115,282)(116,281)(117,280)(118,279)(119,278)(120,277)
(121,271)(122,273)(123,272)(124,275)(125,274)(126,276)(127,291)(128,290)
(129,289)(130,292)(131,294)(132,293)(133,296)(134,295)(135,297)(136,256)
(137,258)(138,257)(139,260)(140,259)(141,261)(142,255)(143,254)(144,253)
(145,252)(146,251)(147,250)(148,244)(149,246)(150,245)(151,248)(152,247)
(153,249)(154,264)(155,263)(156,262)(157,265)(158,267)(159,266)(160,269)
(161,268)(162,270);;
s2 := (  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);;
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope