Polytope of Type {78,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78,6}*1248
if this polytope has a name.
Group : SmallGroup(1248,1438)
Rank : 3
Schlafli Type : {78,6}
Number of vertices, edges, etc : 104, 312, 8
Order of s₀s₁s₂ : 52
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {39,6}*624
   12-fold quotients : {26,2}*104
   13-fold quotients : {6,6}*96
   24-fold quotients : {13,2}*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)(  5, 49)(  6, 50)(  7, 52)(  8, 51)(  9, 45)( 10, 46)( 11, 48)
( 12, 47)( 13, 41)( 14, 42)( 15, 44)( 16, 43)( 17, 37)( 18, 38)( 19, 40)
( 20, 39)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 25, 29)( 26, 30)( 27, 32)
( 28, 31)( 53,105)( 54,106)( 55,108)( 56,107)( 57,153)( 58,154)( 59,156)
( 60,155)( 61,149)( 62,150)( 63,152)( 64,151)( 65,145)( 66,146)( 67,148)
( 68,147)( 69,141)( 70,142)( 71,144)( 72,143)( 73,137)( 74,138)( 75,140)
( 76,139)( 77,133)( 78,134)( 79,136)( 80,135)( 81,129)( 82,130)( 83,132)
( 84,131)( 85,125)( 86,126)( 87,128)( 88,127)( 89,121)( 90,122)( 91,124)
( 92,123)( 93,117)( 94,118)( 95,120)( 96,119)( 97,113)( 98,114)( 99,116)
(100,115)(101,109)(102,110)(103,112)(104,111)(159,160)(161,205)(162,206)
(163,208)(164,207)(165,201)(166,202)(167,204)(168,203)(169,197)(170,198)
(171,200)(172,199)(173,193)(174,194)(175,196)(176,195)(177,189)(178,190)
(179,192)(180,191)(181,185)(182,186)(183,188)(184,187)(209,261)(210,262)
(211,264)(212,263)(213,309)(214,310)(215,312)(216,311)(217,305)(218,306)
(219,308)(220,307)(221,301)(222,302)(223,304)(224,303)(225,297)(226,298)
(227,300)(228,299)(229,293)(230,294)(231,296)(232,295)(233,289)(234,290)
(235,292)(236,291)(237,285)(238,286)(239,288)(240,287)(241,281)(242,282)
(243,284)(244,283)(245,277)(246,278)(247,280)(248,279)(249,273)(250,274)
(251,276)(252,275)(253,269)(254,270)(255,272)(256,271)(257,265)(258,266)
(259,268)(260,267);;
s1 := (  1,213)(  2,216)(  3,215)(  4,214)(  5,209)(  6,212)(  7,211)(  8,210)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,249)( 18,252)( 19,251)( 20,250)( 21,245)( 22,248)( 23,247)( 24,246)
( 25,241)( 26,244)( 27,243)( 28,242)( 29,237)( 30,240)( 31,239)( 32,238)
( 33,233)( 34,236)( 35,235)( 36,234)( 37,229)( 38,232)( 39,231)( 40,230)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,217)( 50,220)( 51,219)( 52,218)( 53,161)( 54,164)( 55,163)( 56,162)
( 57,157)( 58,160)( 59,159)( 60,158)( 61,205)( 62,208)( 63,207)( 64,206)
( 65,201)( 66,204)( 67,203)( 68,202)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)( 80,190)
( 81,185)( 82,188)( 83,187)( 84,186)( 85,181)( 86,184)( 87,183)( 88,182)
( 89,177)( 90,180)( 91,179)( 92,178)( 93,173)( 94,176)( 95,175)( 96,174)
( 97,169)( 98,172)( 99,171)(100,170)(101,165)(102,168)(103,167)(104,166)
(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)(112,262)
(113,309)(114,312)(115,311)(116,310)(117,305)(118,308)(119,307)(120,306)
(121,301)(122,304)(123,303)(124,302)(125,297)(126,300)(127,299)(128,298)
(129,293)(130,296)(131,295)(132,294)(133,289)(134,292)(135,291)(136,290)
(137,285)(138,288)(139,287)(140,286)(141,281)(142,284)(143,283)(144,282)
(145,277)(146,280)(147,279)(148,278)(149,273)(150,276)(151,275)(152,274)
(153,269)(154,272)(155,271)(156,270);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)
( 33, 34)( 37, 38)( 41, 42)( 45, 46)( 49, 50)( 53,106)( 54,105)( 55,107)
( 56,108)( 57,110)( 58,109)( 59,111)( 60,112)( 61,114)( 62,113)( 63,115)
( 64,116)( 65,118)( 66,117)( 67,119)( 68,120)( 69,122)( 70,121)( 71,123)
( 72,124)( 73,126)( 74,125)( 75,127)( 76,128)( 77,130)( 78,129)( 79,131)
( 80,132)( 81,134)( 82,133)( 83,135)( 84,136)( 85,138)( 86,137)( 87,139)
( 88,140)( 89,142)( 90,141)( 91,143)( 92,144)( 93,146)( 94,145)( 95,147)
( 96,148)( 97,150)( 98,149)( 99,151)(100,152)(101,154)(102,153)(103,155)
(104,156)(157,158)(161,162)(165,166)(169,170)(173,174)(177,178)(181,182)
(185,186)(189,190)(193,194)(197,198)(201,202)(205,206)(209,262)(210,261)
(211,263)(212,264)(213,266)(214,265)(215,267)(216,268)(217,270)(218,269)
(219,271)(220,272)(221,274)(222,273)(223,275)(224,276)(225,278)(226,277)
(227,279)(228,280)(229,282)(230,281)(231,283)(232,284)(233,286)(234,285)
(235,287)(236,288)(237,290)(238,289)(239,291)(240,292)(241,294)(242,293)
(243,295)(244,296)(245,298)(246,297)(247,299)(248,300)(249,302)(250,301)
(251,303)(252,304)(253,306)(254,305)(255,307)(256,308)(257,310)(258,309)
(259,311)(260,312);;
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*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*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*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope