Polytope of Type {54,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,12}*1296b
if this polytope has a name.
Group : SmallGroup(1296,904)
Rank : 3
Schlafli Type : {54,12}
Number of vertices, edges, etc : 54, 324, 12
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 6
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 : {54,6}*648b
   3-fold quotients : {54,4}*432a, {18,12}*432b
   4-fold quotients : {27,6}*324
   6-fold quotients : {54,2}*216, {18,6}*216b
   9-fold quotients : {18,4}*144a, {6,12}*144c
   12-fold quotients : {27,2}*108, {9,6}*108
   18-fold quotients : {18,2}*72, {6,6}*72c
   27-fold quotients : {6,4}*48a
   36-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {2,4}*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,  9)(  5,  8)(  6,  7)( 10, 19)( 11, 21)( 12, 20)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 63)( 29, 62)( 30, 61)
( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 81)( 38, 80)
( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)( 46, 72)
( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,100)( 92,102)( 93,101)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(109,144)(110,143)(111,142)
(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,162)(119,161)
(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)
(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145)
(164,165)(166,171)(167,170)(168,169)(172,181)(173,183)(174,182)(175,189)
(176,188)(177,187)(178,186)(179,185)(180,184)(190,225)(191,224)(192,223)
(193,222)(194,221)(195,220)(196,219)(197,218)(198,217)(199,243)(200,242)
(201,241)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)(208,234)
(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)
(245,246)(247,252)(248,251)(249,250)(253,262)(254,264)(255,263)(256,270)
(257,269)(258,268)(259,267)(260,266)(261,265)(271,306)(272,305)(273,304)
(274,303)(275,302)(276,301)(277,300)(278,299)(279,298)(280,324)(281,323)
(282,322)(283,321)(284,320)(285,319)(286,318)(287,317)(288,316)(289,315)
(290,314)(291,313)(292,312)(293,311)(294,310)(295,309)(296,308)(297,307);;
s1 := (  1,199)(  2,201)(  3,200)(  4,207)(  5,206)(  6,205)(  7,204)(  8,203)
(  9,202)( 10,190)( 11,192)( 12,191)( 13,198)( 14,197)( 15,196)( 16,195)
( 17,194)( 18,193)( 19,208)( 20,210)( 21,209)( 22,216)( 23,215)( 24,214)
( 25,213)( 26,212)( 27,211)( 28,172)( 29,174)( 30,173)( 31,180)( 32,179)
( 33,178)( 34,177)( 35,176)( 36,175)( 37,163)( 38,165)( 39,164)( 40,171)
( 41,170)( 42,169)( 43,168)( 44,167)( 45,166)( 46,181)( 47,183)( 48,182)
( 49,189)( 50,188)( 51,187)( 52,186)( 53,185)( 54,184)( 55,234)( 56,233)
( 57,232)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)( 64,225)
( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,218)( 72,217)
( 73,243)( 74,242)( 75,241)( 76,240)( 77,239)( 78,238)( 79,237)( 80,236)
( 81,235)( 82,280)( 83,282)( 84,281)( 85,288)( 86,287)( 87,286)( 88,285)
( 89,284)( 90,283)( 91,271)( 92,273)( 93,272)( 94,279)( 95,278)( 96,277)
( 97,276)( 98,275)( 99,274)(100,289)(101,291)(102,290)(103,297)(104,296)
(105,295)(106,294)(107,293)(108,292)(109,253)(110,255)(111,254)(112,261)
(113,260)(114,259)(115,258)(116,257)(117,256)(118,244)(119,246)(120,245)
(121,252)(122,251)(123,250)(124,249)(125,248)(126,247)(127,262)(128,264)
(129,263)(130,270)(131,269)(132,268)(133,267)(134,266)(135,265)(136,315)
(137,314)(138,313)(139,312)(140,311)(141,310)(142,309)(143,308)(144,307)
(145,306)(146,305)(147,304)(148,303)(149,302)(150,301)(151,300)(152,299)
(153,298)(154,324)(155,323)(156,322)(157,321)(158,320)(159,319)(160,318)
(161,317)(162,316);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,244)(164,245)
(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)(171,252)(172,262)
(173,263)(174,264)(175,265)(176,266)(177,267)(178,268)(179,269)(180,270)
(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)(188,260)
(189,261)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)(196,277)
(197,278)(198,279)(199,289)(200,290)(201,291)(202,292)(203,293)(204,294)
(205,295)(206,296)(207,297)(208,280)(209,281)(210,282)(211,283)(212,284)
(213,285)(214,286)(215,287)(216,288)(217,298)(218,299)(219,300)(220,301)
(221,302)(222,303)(223,304)(224,305)(225,306)(226,316)(227,317)(228,318)
(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)(235,307)(236,308)
(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315);;
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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,  9)(  5,  8)(  6,  7)( 10, 19)( 11, 21)( 12, 20)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 63)( 29, 62)
( 30, 61)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 81)
( 38, 80)( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)
( 46, 72)( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)
( 54, 64)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,100)( 92,102)( 93,101)
( 94,108)( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(109,144)(110,143)
(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,162)
(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)
(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)
(135,145)(164,165)(166,171)(167,170)(168,169)(172,181)(173,183)(174,182)
(175,189)(176,188)(177,187)(178,186)(179,185)(180,184)(190,225)(191,224)
(192,223)(193,222)(194,221)(195,220)(196,219)(197,218)(198,217)(199,243)
(200,242)(201,241)(202,240)(203,239)(204,238)(205,237)(206,236)(207,235)
(208,234)(209,233)(210,232)(211,231)(212,230)(213,229)(214,228)(215,227)
(216,226)(245,246)(247,252)(248,251)(249,250)(253,262)(254,264)(255,263)
(256,270)(257,269)(258,268)(259,267)(260,266)(261,265)(271,306)(272,305)
(273,304)(274,303)(275,302)(276,301)(277,300)(278,299)(279,298)(280,324)
(281,323)(282,322)(283,321)(284,320)(285,319)(286,318)(287,317)(288,316)
(289,315)(290,314)(291,313)(292,312)(293,311)(294,310)(295,309)(296,308)
(297,307);
s1 := Sym(324)!(  1,199)(  2,201)(  3,200)(  4,207)(  5,206)(  6,205)(  7,204)
(  8,203)(  9,202)( 10,190)( 11,192)( 12,191)( 13,198)( 14,197)( 15,196)
( 16,195)( 17,194)( 18,193)( 19,208)( 20,210)( 21,209)( 22,216)( 23,215)
( 24,214)( 25,213)( 26,212)( 27,211)( 28,172)( 29,174)( 30,173)( 31,180)
( 32,179)( 33,178)( 34,177)( 35,176)( 36,175)( 37,163)( 38,165)( 39,164)
( 40,171)( 41,170)( 42,169)( 43,168)( 44,167)( 45,166)( 46,181)( 47,183)
( 48,182)( 49,189)( 50,188)( 51,187)( 52,186)( 53,185)( 54,184)( 55,234)
( 56,233)( 57,232)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)
( 64,225)( 65,224)( 66,223)( 67,222)( 68,221)( 69,220)( 70,219)( 71,218)
( 72,217)( 73,243)( 74,242)( 75,241)( 76,240)( 77,239)( 78,238)( 79,237)
( 80,236)( 81,235)( 82,280)( 83,282)( 84,281)( 85,288)( 86,287)( 87,286)
( 88,285)( 89,284)( 90,283)( 91,271)( 92,273)( 93,272)( 94,279)( 95,278)
( 96,277)( 97,276)( 98,275)( 99,274)(100,289)(101,291)(102,290)(103,297)
(104,296)(105,295)(106,294)(107,293)(108,292)(109,253)(110,255)(111,254)
(112,261)(113,260)(114,259)(115,258)(116,257)(117,256)(118,244)(119,246)
(120,245)(121,252)(122,251)(123,250)(124,249)(125,248)(126,247)(127,262)
(128,264)(129,263)(130,270)(131,269)(132,268)(133,267)(134,266)(135,265)
(136,315)(137,314)(138,313)(139,312)(140,311)(141,310)(142,309)(143,308)
(144,307)(145,306)(146,305)(147,304)(148,303)(149,302)(150,301)(151,300)
(152,299)(153,298)(154,324)(155,323)(156,322)(157,321)(158,320)(159,319)
(160,318)(161,317)(162,316);
s2 := Sym(324)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(163,244)
(164,245)(165,246)(166,247)(167,248)(168,249)(169,250)(170,251)(171,252)
(172,262)(173,263)(174,264)(175,265)(176,266)(177,267)(178,268)(179,269)
(180,270)(181,253)(182,254)(183,255)(184,256)(185,257)(186,258)(187,259)
(188,260)(189,261)(190,271)(191,272)(192,273)(193,274)(194,275)(195,276)
(196,277)(197,278)(198,279)(199,289)(200,290)(201,291)(202,292)(203,293)
(204,294)(205,295)(206,296)(207,297)(208,280)(209,281)(210,282)(211,283)
(212,284)(213,285)(214,286)(215,287)(216,288)(217,298)(218,299)(219,300)
(220,301)(221,302)(222,303)(223,304)(224,305)(225,306)(226,316)(227,317)
(228,318)(229,319)(230,320)(231,321)(232,322)(233,323)(234,324)(235,307)
(236,308)(237,309)(238,310)(239,311)(240,312)(241,313)(242,314)(243,315);
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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