Polytope of Type {12,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,54}*1296b
if this polytope has a name.
Group : SmallGroup(1296,904)
Rank : 3
Schlafli Type : {12,54}
Number of vertices, edges, etc : 12, 324, 54
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 : {6,54}*648b
   3-fold quotients : {4,54}*432a, {12,18}*432b
   4-fold quotients : {6,27}*324
   6-fold quotients : {2,54}*216, {6,18}*216b
   9-fold quotients : {4,18}*144a, {12,6}*144c
   12-fold quotients : {2,27}*108, {6,9}*108
   18-fold quotients : {2,18}*72, {6,6}*72b
   27-fold quotients : {4,6}*48a
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {2,6}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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