Polytope of Type {8,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,78}*1248
Also Known As : {8,78|2}. if this polytope has another name.
Group : SmallGroup(1248,1084)
Rank : 3
Schlafli Type : {8,78}
Number of vertices, edges, etc : 8, 312, 78
Order of s₀s₁s₂ : 312
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 : {4,78}*624a
   3-fold quotients : {8,26}*416
   4-fold quotients : {2,78}*312
   6-fold quotients : {4,26}*208
   8-fold quotients : {2,39}*156
   12-fold quotients : {2,26}*104
   13-fold quotients : {8,6}*96
   24-fold quotients : {2,13}*52
   26-fold quotients : {4,6}*48a
   39-fold quotients : {8,2}*32
   52-fold quotients : {2,6}*24
   78-fold quotients : {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 := ( 79,118)( 80,119)( 81,120)( 82,121)( 83,122)( 84,123)( 85,124)( 86,125)
( 87,126)( 88,127)( 89,128)( 90,129)( 91,130)( 92,131)( 93,132)( 94,133)
( 95,134)( 96,135)( 97,136)( 98,137)( 99,138)(100,139)(101,140)(102,141)
(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)(110,149)
(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)(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)( 79,274)( 80,286)
( 81,285)( 82,284)( 83,283)( 84,282)( 85,281)( 86,280)( 87,279)( 88,278)
( 89,277)( 90,276)( 91,275)( 92,300)( 93,312)( 94,311)( 95,310)( 96,309)
( 97,308)( 98,307)( 99,306)(100,305)(101,304)(102,303)(103,302)(104,301)
(105,287)(106,299)(107,298)(108,297)(109,296)(110,295)(111,294)(112,293)
(113,292)(114,291)(115,290)(116,289)(117,288)(118,235)(119,247)(120,246)
(121,245)(122,244)(123,243)(124,242)(125,241)(126,240)(127,239)(128,238)
(129,237)(130,236)(131,261)(132,273)(133,272)(134,271)(135,270)(136,269)
(137,268)(138,267)(139,266)(140,265)(141,264)(142,263)(143,262)(144,248)
(145,260)(146,259)(147,258)(148,257)(149,256)(150,255)(151,254)(152,253)
(153,252)(154,251)(155,250)(156,249);;
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)(157,171)(158,170)(159,182)(160,181)
(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)
(169,172)(183,184)(185,195)(186,194)(187,193)(188,192)(189,191)(196,210)
(197,209)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)(204,215)
(205,214)(206,213)(207,212)(208,211)(222,223)(224,234)(225,233)(226,232)
(227,231)(228,230)(235,249)(236,248)(237,260)(238,259)(239,258)(240,257)
(241,256)(242,255)(243,254)(244,253)(245,252)(246,251)(247,250)(261,262)
(263,273)(264,272)(265,271)(266,270)(267,269)(274,288)(275,287)(276,299)
(277,298)(278,297)(279,296)(280,295)(281,294)(282,293)(283,292)(284,291)
(285,290)(286,289)(300,301)(302,312)(303,311)(304,310)(305,309)(306,308);;
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, 
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 79,118)( 80,119)( 81,120)( 82,121)( 83,122)( 84,123)( 85,124)
( 86,125)( 87,126)( 88,127)( 89,128)( 90,129)( 91,130)( 92,131)( 93,132)
( 94,133)( 95,134)( 96,135)( 97,136)( 98,137)( 99,138)(100,139)(101,140)
(102,141)(103,142)(104,143)(105,144)(106,145)(107,146)(108,147)(109,148)
(110,149)(111,150)(112,151)(113,152)(114,153)(115,154)(116,155)(117,156)
(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)( 79,274)
( 80,286)( 81,285)( 82,284)( 83,283)( 84,282)( 85,281)( 86,280)( 87,279)
( 88,278)( 89,277)( 90,276)( 91,275)( 92,300)( 93,312)( 94,311)( 95,310)
( 96,309)( 97,308)( 98,307)( 99,306)(100,305)(101,304)(102,303)(103,302)
(104,301)(105,287)(106,299)(107,298)(108,297)(109,296)(110,295)(111,294)
(112,293)(113,292)(114,291)(115,290)(116,289)(117,288)(118,235)(119,247)
(120,246)(121,245)(122,244)(123,243)(124,242)(125,241)(126,240)(127,239)
(128,238)(129,237)(130,236)(131,261)(132,273)(133,272)(134,271)(135,270)
(136,269)(137,268)(138,267)(139,266)(140,265)(141,264)(142,263)(143,262)
(144,248)(145,260)(146,259)(147,258)(148,257)(149,256)(150,255)(151,254)
(152,253)(153,252)(154,251)(155,250)(156,249);
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)(157,171)(158,170)(159,182)
(160,181)(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)
(168,173)(169,172)(183,184)(185,195)(186,194)(187,193)(188,192)(189,191)
(196,210)(197,209)(198,221)(199,220)(200,219)(201,218)(202,217)(203,216)
(204,215)(205,214)(206,213)(207,212)(208,211)(222,223)(224,234)(225,233)
(226,232)(227,231)(228,230)(235,249)(236,248)(237,260)(238,259)(239,258)
(240,257)(241,256)(242,255)(243,254)(244,253)(245,252)(246,251)(247,250)
(261,262)(263,273)(264,272)(265,271)(266,270)(267,269)(274,288)(275,287)
(276,299)(277,298)(278,297)(279,296)(280,295)(281,294)(282,293)(283,292)
(284,291)(285,290)(286,289)(300,301)(302,312)(303,311)(304,310)(305,309)
(306,308);
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, 
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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