Polytope of Type {54,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,12}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1781)
Rank : 3
Schlafli Type : {54,12}
Number of vertices, edges, etc : 54, 324, 12
Order of s₀s₁s₂ : 27
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
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) :
   3-fold quotients : {54,4}*432c, {18,12}*432c
   6-fold quotients : {27,4}*216
   9-fold quotients : {18,4}*144c, {6,12}*144d
   18-fold quotients : {9,4}*72
   27-fold quotients : {6,4}*48b
   54-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope