Polytope of Type {6,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,78}*1248
if this polytope has a name.
Group : SmallGroup(1248,1438)
Rank : 3
Schlafli Type : {6,78}
Number of vertices, edges, etc : 8, 312, 104
Order of s₀s₁s₂ : 52
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,39}*624
   12-fold quotients : {2,26}*104
   13-fold quotients : {6,6}*96
   24-fold quotients : {2,13}*52
   26-fold quotients : {3,6}*48, {6,3}*48
   52-fold quotients : {3,3}*24
   156-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope