Polytope of Type {78,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78,8}*1248
Also Known As : {78,8|2}. if this polytope has another name.
Group : SmallGroup(1248,1084)
Rank : 3
Schlafli Type : {78,8}
Number of vertices, edges, etc : 78, 312, 8
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 : {78,4}*624a
3-fold quotients : {26,8}*416
4-fold quotients : {78,2}*312
6-fold quotients : {26,4}*208
8-fold quotients : {39,2}*156
12-fold quotients : {26,2}*104
13-fold quotients : {6,8}*96
24-fold quotients : {13,2}*52
26-fold quotients : {6,4}*48a
39-fold quotients : {2,8}*32
52-fold quotients : {6,2}*24
78-fold quotients : {2,4}*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)(158,169)(159,168)(160,167)(161,166)
(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)(175,191)
(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(197,208)
(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)(210,234)(211,233)
(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)(218,226)(219,225)
(220,224)(221,223)(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)
(248,261)(249,273)(250,272)(251,271)(252,270)(253,269)(254,268)(255,267)
(256,266)(257,265)(258,264)(259,263)(260,262)(275,286)(276,285)(277,284)
(278,283)(279,282)(280,281)(287,300)(288,312)(289,311)(290,310)(291,309)
(292,308)(293,307)(294,306)(295,305)(296,304)(297,303)(298,302)(299,301);;
s1 := ( 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)( 79,132)( 80,131)
( 81,143)( 82,142)( 83,141)( 84,140)( 85,139)( 86,138)( 87,137)( 88,136)
( 89,135)( 90,134)( 91,133)( 92,119)( 93,118)( 94,130)( 95,129)( 96,128)
( 97,127)( 98,126)( 99,125)(100,124)(101,123)(102,122)(103,121)(104,120)
(105,145)(106,144)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)
(113,150)(114,149)(115,148)(116,147)(117,146)(157,249)(158,248)(159,260)
(160,259)(161,258)(162,257)(163,256)(164,255)(165,254)(166,253)(167,252)
(168,251)(169,250)(170,236)(171,235)(172,247)(173,246)(174,245)(175,244)
(176,243)(177,242)(178,241)(179,240)(180,239)(181,238)(182,237)(183,262)
(184,261)(185,273)(186,272)(187,271)(188,270)(189,269)(190,268)(191,267)
(192,266)(193,265)(194,264)(195,263)(196,288)(197,287)(198,299)(199,298)
(200,297)(201,296)(202,295)(203,294)(204,293)(205,292)(206,291)(207,290)
(208,289)(209,275)(210,274)(211,286)(212,285)(213,284)(214,283)(215,282)
(216,281)(217,280)(218,279)(219,278)(220,277)(221,276)(222,301)(223,300)
(224,312)(225,311)(226,310)(227,309)(228,308)(229,307)(230,306)(231,305)
(232,304)(233,303)(234,302);;
s2 := ( 1,157)( 2,158)( 3,159)( 4,160)( 5,161)( 6,162)( 7,163)( 8,164)
( 9,165)( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)( 16,172)
( 17,173)( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)( 24,180)
( 25,181)( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)( 32,188)
( 33,189)( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)( 40,196)
( 41,197)( 42,198)( 43,199)( 44,200)( 45,201)( 46,202)( 47,203)( 48,204)
( 49,205)( 50,206)( 51,207)( 52,208)( 53,209)( 54,210)( 55,211)( 56,212)
( 57,213)( 58,214)( 59,215)( 60,216)( 61,217)( 62,218)( 63,219)( 64,220)
( 65,221)( 66,222)( 67,223)( 68,224)( 69,225)( 70,226)( 71,227)( 72,228)
( 73,229)( 74,230)( 75,231)( 76,232)( 77,233)( 78,234)( 79,274)( 80,275)
( 81,276)( 82,277)( 83,278)( 84,279)( 85,280)( 86,281)( 87,282)( 88,283)
( 89,284)( 90,285)( 91,286)( 92,287)( 93,288)( 94,289)( 95,290)( 96,291)
( 97,292)( 98,293)( 99,294)(100,295)(101,296)(102,297)(103,298)(104,299)
(105,300)(106,301)(107,302)(108,303)(109,304)(110,305)(111,306)(112,307)
(113,308)(114,309)(115,310)(116,311)(117,312)(118,235)(119,236)(120,237)
(121,238)(122,239)(123,240)(124,241)(125,242)(126,243)(127,244)(128,245)
(129,246)(130,247)(131,248)(132,249)(133,250)(134,251)(135,252)(136,253)
(137,254)(138,255)(139,256)(140,257)(141,258)(142,259)(143,260)(144,261)
(145,262)(146,263)(147,264)(148,265)(149,266)(150,267)(151,268)(152,269)
(153,270)(154,271)(155,272)(156,273);;
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*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 ];;
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)(158,169)(159,168)(160,167)
(161,166)(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)
(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)
(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)(210,234)
(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)(218,226)
(219,225)(220,224)(221,223)(236,247)(237,246)(238,245)(239,244)(240,243)
(241,242)(248,261)(249,273)(250,272)(251,271)(252,270)(253,269)(254,268)
(255,267)(256,266)(257,265)(258,264)(259,263)(260,262)(275,286)(276,285)
(277,284)(278,283)(279,282)(280,281)(287,300)(288,312)(289,311)(290,310)
(291,309)(292,308)(293,307)(294,306)(295,305)(296,304)(297,303)(298,302)
(299,301);
s1 := 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)( 79,132)
( 80,131)( 81,143)( 82,142)( 83,141)( 84,140)( 85,139)( 86,138)( 87,137)
( 88,136)( 89,135)( 90,134)( 91,133)( 92,119)( 93,118)( 94,130)( 95,129)
( 96,128)( 97,127)( 98,126)( 99,125)(100,124)(101,123)(102,122)(103,121)
(104,120)(105,145)(106,144)(107,156)(108,155)(109,154)(110,153)(111,152)
(112,151)(113,150)(114,149)(115,148)(116,147)(117,146)(157,249)(158,248)
(159,260)(160,259)(161,258)(162,257)(163,256)(164,255)(165,254)(166,253)
(167,252)(168,251)(169,250)(170,236)(171,235)(172,247)(173,246)(174,245)
(175,244)(176,243)(177,242)(178,241)(179,240)(180,239)(181,238)(182,237)
(183,262)(184,261)(185,273)(186,272)(187,271)(188,270)(189,269)(190,268)
(191,267)(192,266)(193,265)(194,264)(195,263)(196,288)(197,287)(198,299)
(199,298)(200,297)(201,296)(202,295)(203,294)(204,293)(205,292)(206,291)
(207,290)(208,289)(209,275)(210,274)(211,286)(212,285)(213,284)(214,283)
(215,282)(216,281)(217,280)(218,279)(219,278)(220,277)(221,276)(222,301)
(223,300)(224,312)(225,311)(226,310)(227,309)(228,308)(229,307)(230,306)
(231,305)(232,304)(233,303)(234,302);
s2 := Sym(312)!( 1,157)( 2,158)( 3,159)( 4,160)( 5,161)( 6,162)( 7,163)
( 8,164)( 9,165)( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)
( 16,172)( 17,173)( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)
( 24,180)( 25,181)( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)
( 32,188)( 33,189)( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)
( 40,196)( 41,197)( 42,198)( 43,199)( 44,200)( 45,201)( 46,202)( 47,203)
( 48,204)( 49,205)( 50,206)( 51,207)( 52,208)( 53,209)( 54,210)( 55,211)
( 56,212)( 57,213)( 58,214)( 59,215)( 60,216)( 61,217)( 62,218)( 63,219)
( 64,220)( 65,221)( 66,222)( 67,223)( 68,224)( 69,225)( 70,226)( 71,227)
( 72,228)( 73,229)( 74,230)( 75,231)( 76,232)( 77,233)( 78,234)( 79,274)
( 80,275)( 81,276)( 82,277)( 83,278)( 84,279)( 85,280)( 86,281)( 87,282)
( 88,283)( 89,284)( 90,285)( 91,286)( 92,287)( 93,288)( 94,289)( 95,290)
( 96,291)( 97,292)( 98,293)( 99,294)(100,295)(101,296)(102,297)(103,298)
(104,299)(105,300)(106,301)(107,302)(108,303)(109,304)(110,305)(111,306)
(112,307)(113,308)(114,309)(115,310)(116,311)(117,312)(118,235)(119,236)
(120,237)(121,238)(122,239)(123,240)(124,241)(125,242)(126,243)(127,244)
(128,245)(129,246)(130,247)(131,248)(132,249)(133,250)(134,251)(135,252)
(136,253)(137,254)(138,255)(139,256)(140,257)(141,258)(142,259)(143,260)
(144,261)(145,262)(146,263)(147,264)(148,265)(149,266)(150,267)(151,268)
(152,269)(153,270)(154,271)(155,272)(156,273);
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*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 >;
References : None.
to this polytope