Polytope of Type {156,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {156,4}*1248a
Also Known As : {156,4|2}. if this polytope has another name.
Group : SmallGroup(1248,1057)
Rank : 3
Schlafli Type : {156,4}
Number of vertices, edges, etc : 156, 312, 4
Order of s₀s₁s₂ : 156
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {156,2}*624, {78,4}*624a
   3-fold quotients : {52,4}*416
   4-fold quotients : {78,2}*312
   6-fold quotients : {52,2}*208, {26,4}*208
   8-fold quotients : {39,2}*156
   12-fold quotients : {26,2}*104
   13-fold quotients : {12,4}*96a
   24-fold quotients : {13,2}*52
   26-fold quotients : {12,2}*48, {6,4}*48a
   39-fold quotients : {4,4}*32
   52-fold quotients : {6,2}*24
   78-fold quotients : {2,4}*16, {4,2}*16
   104-fold quotients : {3,2}*12
   156-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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