Polytope of Type {54,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,12}*1296a
Also Known As : {54,12|2}. if this polytope has another name.
Group : SmallGroup(1296,825)
Rank : 3
Schlafli Type : {54,12}
Number of vertices, edges, etc : 54, 324, 12
Order of s0s1s2 : 108
Order of s0s1s2s1 : 2
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}*648a
3-fold quotients : {54,4}*432a, {18,12}*432a
6-fold quotients : {54,2}*216, {18,6}*216a
9-fold quotients : {18,4}*144a, {6,12}*144a
12-fold quotients : {27,2}*108
18-fold quotients : {18,2}*72, {6,6}*72a
27-fold quotients : {2,12}*48, {6,4}*48a
36-fold quotients : {9,2}*36
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 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)
( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 63)( 29, 62)( 30, 61)( 31, 60)
( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 72)( 38, 71)( 39, 70)
( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 81)( 47, 80)
( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)( 83, 84)
( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)
(103,108)(104,107)(105,106)(109,144)(110,143)(111,142)(112,141)(113,140)
(114,139)(115,138)(116,137)(117,136)(118,153)(119,152)(120,151)(121,150)
(122,149)(123,148)(124,147)(125,146)(126,145)(127,162)(128,161)(129,160)
(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(164,165)(166,171)
(167,170)(168,169)(173,174)(175,180)(176,179)(177,178)(182,183)(184,189)
(185,188)(186,187)(190,225)(191,224)(192,223)(193,222)(194,221)(195,220)
(196,219)(197,218)(198,217)(199,234)(200,233)(201,232)(202,231)(203,230)
(204,229)(205,228)(206,227)(207,226)(208,243)(209,242)(210,241)(211,240)
(212,239)(213,238)(214,237)(215,236)(216,235)(245,246)(247,252)(248,251)
(249,250)(254,255)(256,261)(257,260)(258,259)(263,264)(265,270)(266,269)
(267,268)(271,306)(272,305)(273,304)(274,303)(275,302)(276,301)(277,300)
(278,299)(279,298)(280,315)(281,314)(282,313)(283,312)(284,311)(285,310)
(286,309)(287,308)(288,307)(289,324)(290,323)(291,322)(292,321)(293,320)
(294,319)(295,318)(296,317)(297,316);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)
( 9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)( 89,113)
( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)( 97,132)
( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)(105,124)
(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)(145,162)
(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)
(163,271)(164,273)(165,272)(166,279)(167,278)(168,277)(169,276)(170,275)
(171,274)(172,289)(173,291)(174,290)(175,297)(176,296)(177,295)(178,294)
(179,293)(180,292)(181,280)(182,282)(183,281)(184,288)(185,287)(186,286)
(187,285)(188,284)(189,283)(190,244)(191,246)(192,245)(193,252)(194,251)
(195,250)(196,249)(197,248)(198,247)(199,262)(200,264)(201,263)(202,270)
(203,269)(204,268)(205,267)(206,266)(207,265)(208,253)(209,255)(210,254)
(211,261)(212,260)(213,259)(214,258)(215,257)(216,256)(217,306)(218,305)
(219,304)(220,303)(221,302)(222,301)(223,300)(224,299)(225,298)(226,324)
(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)(234,316)
(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)(242,308)
(243,307);;
s2 := ( 1,172)( 2,173)( 3,174)( 4,175)( 5,176)( 6,177)( 7,178)( 8,179)
( 9,180)( 10,163)( 11,164)( 12,165)( 13,166)( 14,167)( 15,168)( 16,169)
( 17,170)( 18,171)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)
( 25,187)( 26,188)( 27,189)( 28,199)( 29,200)( 30,201)( 31,202)( 32,203)
( 33,204)( 34,205)( 35,206)( 36,207)( 37,190)( 38,191)( 39,192)( 40,193)
( 41,194)( 42,195)( 43,196)( 44,197)( 45,198)( 46,208)( 47,209)( 48,210)
( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,226)( 56,227)
( 57,228)( 58,229)( 59,230)( 60,231)( 61,232)( 62,233)( 63,234)( 64,217)
( 65,218)( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,224)( 72,225)
( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)
( 81,243)( 82,253)( 83,254)( 84,255)( 85,256)( 86,257)( 87,258)( 88,259)
( 89,260)( 90,261)( 91,244)( 92,245)( 93,246)( 94,247)( 95,248)( 96,249)
( 97,250)( 98,251)( 99,252)(100,262)(101,263)(102,264)(103,265)(104,266)
(105,267)(106,268)(107,269)(108,270)(109,280)(110,281)(111,282)(112,283)
(113,284)(114,285)(115,286)(116,287)(117,288)(118,271)(119,272)(120,273)
(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,289)(128,290)
(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,307)
(137,308)(138,309)(139,310)(140,311)(141,312)(142,313)(143,314)(144,315)
(145,298)(146,299)(147,300)(148,301)(149,302)(150,303)(151,304)(152,305)
(153,306)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)
(161,323)(162,324);;
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, 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,
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)( 11, 12)( 13, 18)( 14, 17)
( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 63)( 29, 62)( 30, 61)
( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 72)( 38, 71)
( 39, 70)( 40, 69)( 41, 68)( 42, 67)( 43, 66)( 44, 65)( 45, 64)( 46, 81)
( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)
(101,102)(103,108)(104,107)(105,106)(109,144)(110,143)(111,142)(112,141)
(113,140)(114,139)(115,138)(116,137)(117,136)(118,153)(119,152)(120,151)
(121,150)(122,149)(123,148)(124,147)(125,146)(126,145)(127,162)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(164,165)
(166,171)(167,170)(168,169)(173,174)(175,180)(176,179)(177,178)(182,183)
(184,189)(185,188)(186,187)(190,225)(191,224)(192,223)(193,222)(194,221)
(195,220)(196,219)(197,218)(198,217)(199,234)(200,233)(201,232)(202,231)
(203,230)(204,229)(205,228)(206,227)(207,226)(208,243)(209,242)(210,241)
(211,240)(212,239)(213,238)(214,237)(215,236)(216,235)(245,246)(247,252)
(248,251)(249,250)(254,255)(256,261)(257,260)(258,259)(263,264)(265,270)
(266,269)(267,268)(271,306)(272,305)(273,304)(274,303)(275,302)(276,301)
(277,300)(278,299)(279,298)(280,315)(281,314)(282,313)(283,312)(284,311)
(285,310)(286,309)(287,308)(288,307)(289,324)(290,323)(291,322)(292,321)
(293,320)(294,319)(295,318)(296,317)(297,316);
s1 := Sym(324)!( 1, 28)( 2, 30)( 3, 29)( 4, 36)( 5, 35)( 6, 34)( 7, 33)
( 8, 32)( 9, 31)( 10, 46)( 11, 48)( 12, 47)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 37)( 20, 39)( 21, 38)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 63)( 56, 62)( 57, 61)( 58, 60)
( 64, 81)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)
( 72, 73)( 82,109)( 83,111)( 84,110)( 85,117)( 86,116)( 87,115)( 88,114)
( 89,113)( 90,112)( 91,127)( 92,129)( 93,128)( 94,135)( 95,134)( 96,133)
( 97,132)( 98,131)( 99,130)(100,118)(101,120)(102,119)(103,126)(104,125)
(105,124)(106,123)(107,122)(108,121)(136,144)(137,143)(138,142)(139,141)
(145,162)(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)
(153,154)(163,271)(164,273)(165,272)(166,279)(167,278)(168,277)(169,276)
(170,275)(171,274)(172,289)(173,291)(174,290)(175,297)(176,296)(177,295)
(178,294)(179,293)(180,292)(181,280)(182,282)(183,281)(184,288)(185,287)
(186,286)(187,285)(188,284)(189,283)(190,244)(191,246)(192,245)(193,252)
(194,251)(195,250)(196,249)(197,248)(198,247)(199,262)(200,264)(201,263)
(202,270)(203,269)(204,268)(205,267)(206,266)(207,265)(208,253)(209,255)
(210,254)(211,261)(212,260)(213,259)(214,258)(215,257)(216,256)(217,306)
(218,305)(219,304)(220,303)(221,302)(222,301)(223,300)(224,299)(225,298)
(226,324)(227,323)(228,322)(229,321)(230,320)(231,319)(232,318)(233,317)
(234,316)(235,315)(236,314)(237,313)(238,312)(239,311)(240,310)(241,309)
(242,308)(243,307);
s2 := Sym(324)!( 1,172)( 2,173)( 3,174)( 4,175)( 5,176)( 6,177)( 7,178)
( 8,179)( 9,180)( 10,163)( 11,164)( 12,165)( 13,166)( 14,167)( 15,168)
( 16,169)( 17,170)( 18,171)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)
( 24,186)( 25,187)( 26,188)( 27,189)( 28,199)( 29,200)( 30,201)( 31,202)
( 32,203)( 33,204)( 34,205)( 35,206)( 36,207)( 37,190)( 38,191)( 39,192)
( 40,193)( 41,194)( 42,195)( 43,196)( 44,197)( 45,198)( 46,208)( 47,209)
( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,226)
( 56,227)( 57,228)( 58,229)( 59,230)( 60,231)( 61,232)( 62,233)( 63,234)
( 64,217)( 65,218)( 66,219)( 67,220)( 68,221)( 69,222)( 70,223)( 71,224)
( 72,225)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)
( 80,242)( 81,243)( 82,253)( 83,254)( 84,255)( 85,256)( 86,257)( 87,258)
( 88,259)( 89,260)( 90,261)( 91,244)( 92,245)( 93,246)( 94,247)( 95,248)
( 96,249)( 97,250)( 98,251)( 99,252)(100,262)(101,263)(102,264)(103,265)
(104,266)(105,267)(106,268)(107,269)(108,270)(109,280)(110,281)(111,282)
(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,271)(119,272)
(120,273)(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,289)
(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)
(136,307)(137,308)(138,309)(139,310)(140,311)(141,312)(142,313)(143,314)
(144,315)(145,298)(146,299)(147,300)(148,301)(149,302)(150,303)(151,304)
(152,305)(153,306)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)
(160,322)(161,323)(162,324);
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, 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,
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