Polytope of Type {4,156}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,156}*1248a
Also Known As : {4,156|2}. if this polytope has another name.
Group : SmallGroup(1248,1057)
Rank : 3
Schlafli Type : {4,156}
Number of vertices, edges, etc : 4, 312, 156
Order of s₀s₁s₂ : 156
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 : {2,156}*624, {4,78}*624a
   3-fold quotients : {4,52}*416
   4-fold quotients : {2,78}*312
   6-fold quotients : {2,52}*208, {4,26}*208
   8-fold quotients : {2,39}*156
   12-fold quotients : {2,26}*104
   13-fold quotients : {4,12}*96a
   24-fold quotients : {2,13}*52
   26-fold quotients : {2,12}*48, {4,6}*48a
   39-fold quotients : {4,4}*32
   52-fold quotients : {2,6}*24
   78-fold quotients : {2,4}*16, {4,2}*16
   104-fold quotients : {2,3}*12
   156-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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