Polytope of Type {4,78}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,78}*624c
if this polytope has a name.
Group : SmallGroup(624,245)
Rank : 3
Schlafli Type : {4,78}
Number of vertices, edges, etc : 4, 156, 78
Order of s0s1s2 : 39
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,78,2} of size 1248
Vertex Figure Of :
{2,4,78} of size 1248
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,39}*312
13-fold quotients : {4,6}*48b
26-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,78}*1248
3-fold covers : {4,234}*1872c, {12,78}*1872d
Permutation Representation (GAP) :
s0 := ( 1,159)( 2,160)( 3,157)( 4,158)( 5,163)( 6,164)( 7,161)( 8,162)
( 9,167)( 10,168)( 11,165)( 12,166)( 13,171)( 14,172)( 15,169)( 16,170)
( 17,175)( 18,176)( 19,173)( 20,174)( 21,179)( 22,180)( 23,177)( 24,178)
( 25,183)( 26,184)( 27,181)( 28,182)( 29,187)( 30,188)( 31,185)( 32,186)
( 33,191)( 34,192)( 35,189)( 36,190)( 37,195)( 38,196)( 39,193)( 40,194)
( 41,199)( 42,200)( 43,197)( 44,198)( 45,203)( 46,204)( 47,201)( 48,202)
( 49,207)( 50,208)( 51,205)( 52,206)( 53,211)( 54,212)( 55,209)( 56,210)
( 57,215)( 58,216)( 59,213)( 60,214)( 61,219)( 62,220)( 63,217)( 64,218)
( 65,223)( 66,224)( 67,221)( 68,222)( 69,227)( 70,228)( 71,225)( 72,226)
( 73,231)( 74,232)( 75,229)( 76,230)( 77,235)( 78,236)( 79,233)( 80,234)
( 81,239)( 82,240)( 83,237)( 84,238)( 85,243)( 86,244)( 87,241)( 88,242)
( 89,247)( 90,248)( 91,245)( 92,246)( 93,251)( 94,252)( 95,249)( 96,250)
( 97,255)( 98,256)( 99,253)(100,254)(101,259)(102,260)(103,257)(104,258)
(105,263)(106,264)(107,261)(108,262)(109,267)(110,268)(111,265)(112,266)
(113,271)(114,272)(115,269)(116,270)(117,275)(118,276)(119,273)(120,274)
(121,279)(122,280)(123,277)(124,278)(125,283)(126,284)(127,281)(128,282)
(129,287)(130,288)(131,285)(132,286)(133,291)(134,292)(135,289)(136,290)
(137,295)(138,296)(139,293)(140,294)(141,299)(142,300)(143,297)(144,298)
(145,303)(146,304)(147,301)(148,302)(149,307)(150,308)(151,305)(152,306)
(153,311)(154,312)(155,309)(156,310);;
s1 := ( 2, 3)( 5, 49)( 6, 51)( 7, 50)( 8, 52)( 9, 45)( 10, 47)( 11, 46)
( 12, 48)( 13, 41)( 14, 43)( 15, 42)( 16, 44)( 17, 37)( 18, 39)( 19, 38)
( 20, 40)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 25, 29)( 26, 31)( 27, 30)
( 28, 32)( 53,105)( 54,107)( 55,106)( 56,108)( 57,153)( 58,155)( 59,154)
( 60,156)( 61,149)( 62,151)( 63,150)( 64,152)( 65,145)( 66,147)( 67,146)
( 68,148)( 69,141)( 70,143)( 71,142)( 72,144)( 73,137)( 74,139)( 75,138)
( 76,140)( 77,133)( 78,135)( 79,134)( 80,136)( 81,129)( 82,131)( 83,130)
( 84,132)( 85,125)( 86,127)( 87,126)( 88,128)( 89,121)( 90,123)( 91,122)
( 92,124)( 93,117)( 94,119)( 95,118)( 96,120)( 97,113)( 98,115)( 99,114)
(100,116)(101,109)(102,111)(103,110)(104,112)(158,159)(161,205)(162,207)
(163,206)(164,208)(165,201)(166,203)(167,202)(168,204)(169,197)(170,199)
(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,189)(178,191)
(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(209,261)(210,263)
(211,262)(212,264)(213,309)(214,311)(215,310)(216,312)(217,305)(218,307)
(219,306)(220,308)(221,301)(222,303)(223,302)(224,304)(225,297)(226,299)
(227,298)(228,300)(229,293)(230,295)(231,294)(232,296)(233,289)(234,291)
(235,290)(236,292)(237,285)(238,287)(239,286)(240,288)(241,281)(242,283)
(243,282)(244,284)(245,277)(246,279)(247,278)(248,280)(249,273)(250,275)
(251,274)(252,276)(253,269)(254,271)(255,270)(256,272)(257,265)(258,267)
(259,266)(260,268);;
s2 := ( 1,265)( 2,268)( 3,267)( 4,266)( 5,261)( 6,264)( 7,263)( 8,262)
( 9,309)( 10,312)( 11,311)( 12,310)( 13,305)( 14,308)( 15,307)( 16,306)
( 17,301)( 18,304)( 19,303)( 20,302)( 21,297)( 22,300)( 23,299)( 24,298)
( 25,293)( 26,296)( 27,295)( 28,294)( 29,289)( 30,292)( 31,291)( 32,290)
( 33,285)( 34,288)( 35,287)( 36,286)( 37,281)( 38,284)( 39,283)( 40,282)
( 41,277)( 42,280)( 43,279)( 44,278)( 45,273)( 46,276)( 47,275)( 48,274)
( 49,269)( 50,272)( 51,271)( 52,270)( 53,213)( 54,216)( 55,215)( 56,214)
( 57,209)( 58,212)( 59,211)( 60,210)( 61,257)( 62,260)( 63,259)( 64,258)
( 65,253)( 66,256)( 67,255)( 68,254)( 69,249)( 70,252)( 71,251)( 72,250)
( 73,245)( 74,248)( 75,247)( 76,246)( 77,241)( 78,244)( 79,243)( 80,242)
( 81,237)( 82,240)( 83,239)( 84,238)( 85,233)( 86,236)( 87,235)( 88,234)
( 89,229)( 90,232)( 91,231)( 92,230)( 93,225)( 94,228)( 95,227)( 96,226)
( 97,221)( 98,224)( 99,223)(100,222)(101,217)(102,220)(103,219)(104,218)
(105,161)(106,164)(107,163)(108,162)(109,157)(110,160)(111,159)(112,158)
(113,205)(114,208)(115,207)(116,206)(117,201)(118,204)(119,203)(120,202)
(121,197)(122,200)(123,199)(124,198)(125,193)(126,196)(127,195)(128,194)
(129,189)(130,192)(131,191)(132,190)(133,185)(134,188)(135,187)(136,186)
(137,181)(138,184)(139,183)(140,182)(141,177)(142,180)(143,179)(144,178)
(145,173)(146,176)(147,175)(148,174)(149,169)(150,172)(151,171)(152,170)
(153,165)(154,168)(155,167)(156,166);;
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*s2*s1*s0*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 1,159)( 2,160)( 3,157)( 4,158)( 5,163)( 6,164)( 7,161)
( 8,162)( 9,167)( 10,168)( 11,165)( 12,166)( 13,171)( 14,172)( 15,169)
( 16,170)( 17,175)( 18,176)( 19,173)( 20,174)( 21,179)( 22,180)( 23,177)
( 24,178)( 25,183)( 26,184)( 27,181)( 28,182)( 29,187)( 30,188)( 31,185)
( 32,186)( 33,191)( 34,192)( 35,189)( 36,190)( 37,195)( 38,196)( 39,193)
( 40,194)( 41,199)( 42,200)( 43,197)( 44,198)( 45,203)( 46,204)( 47,201)
( 48,202)( 49,207)( 50,208)( 51,205)( 52,206)( 53,211)( 54,212)( 55,209)
( 56,210)( 57,215)( 58,216)( 59,213)( 60,214)( 61,219)( 62,220)( 63,217)
( 64,218)( 65,223)( 66,224)( 67,221)( 68,222)( 69,227)( 70,228)( 71,225)
( 72,226)( 73,231)( 74,232)( 75,229)( 76,230)( 77,235)( 78,236)( 79,233)
( 80,234)( 81,239)( 82,240)( 83,237)( 84,238)( 85,243)( 86,244)( 87,241)
( 88,242)( 89,247)( 90,248)( 91,245)( 92,246)( 93,251)( 94,252)( 95,249)
( 96,250)( 97,255)( 98,256)( 99,253)(100,254)(101,259)(102,260)(103,257)
(104,258)(105,263)(106,264)(107,261)(108,262)(109,267)(110,268)(111,265)
(112,266)(113,271)(114,272)(115,269)(116,270)(117,275)(118,276)(119,273)
(120,274)(121,279)(122,280)(123,277)(124,278)(125,283)(126,284)(127,281)
(128,282)(129,287)(130,288)(131,285)(132,286)(133,291)(134,292)(135,289)
(136,290)(137,295)(138,296)(139,293)(140,294)(141,299)(142,300)(143,297)
(144,298)(145,303)(146,304)(147,301)(148,302)(149,307)(150,308)(151,305)
(152,306)(153,311)(154,312)(155,309)(156,310);
s1 := Sym(312)!( 2, 3)( 5, 49)( 6, 51)( 7, 50)( 8, 52)( 9, 45)( 10, 47)
( 11, 46)( 12, 48)( 13, 41)( 14, 43)( 15, 42)( 16, 44)( 17, 37)( 18, 39)
( 19, 38)( 20, 40)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 25, 29)( 26, 31)
( 27, 30)( 28, 32)( 53,105)( 54,107)( 55,106)( 56,108)( 57,153)( 58,155)
( 59,154)( 60,156)( 61,149)( 62,151)( 63,150)( 64,152)( 65,145)( 66,147)
( 67,146)( 68,148)( 69,141)( 70,143)( 71,142)( 72,144)( 73,137)( 74,139)
( 75,138)( 76,140)( 77,133)( 78,135)( 79,134)( 80,136)( 81,129)( 82,131)
( 83,130)( 84,132)( 85,125)( 86,127)( 87,126)( 88,128)( 89,121)( 90,123)
( 91,122)( 92,124)( 93,117)( 94,119)( 95,118)( 96,120)( 97,113)( 98,115)
( 99,114)(100,116)(101,109)(102,111)(103,110)(104,112)(158,159)(161,205)
(162,207)(163,206)(164,208)(165,201)(166,203)(167,202)(168,204)(169,197)
(170,199)(171,198)(172,200)(173,193)(174,195)(175,194)(176,196)(177,189)
(178,191)(179,190)(180,192)(181,185)(182,187)(183,186)(184,188)(209,261)
(210,263)(211,262)(212,264)(213,309)(214,311)(215,310)(216,312)(217,305)
(218,307)(219,306)(220,308)(221,301)(222,303)(223,302)(224,304)(225,297)
(226,299)(227,298)(228,300)(229,293)(230,295)(231,294)(232,296)(233,289)
(234,291)(235,290)(236,292)(237,285)(238,287)(239,286)(240,288)(241,281)
(242,283)(243,282)(244,284)(245,277)(246,279)(247,278)(248,280)(249,273)
(250,275)(251,274)(252,276)(253,269)(254,271)(255,270)(256,272)(257,265)
(258,267)(259,266)(260,268);
s2 := Sym(312)!( 1,265)( 2,268)( 3,267)( 4,266)( 5,261)( 6,264)( 7,263)
( 8,262)( 9,309)( 10,312)( 11,311)( 12,310)( 13,305)( 14,308)( 15,307)
( 16,306)( 17,301)( 18,304)( 19,303)( 20,302)( 21,297)( 22,300)( 23,299)
( 24,298)( 25,293)( 26,296)( 27,295)( 28,294)( 29,289)( 30,292)( 31,291)
( 32,290)( 33,285)( 34,288)( 35,287)( 36,286)( 37,281)( 38,284)( 39,283)
( 40,282)( 41,277)( 42,280)( 43,279)( 44,278)( 45,273)( 46,276)( 47,275)
( 48,274)( 49,269)( 50,272)( 51,271)( 52,270)( 53,213)( 54,216)( 55,215)
( 56,214)( 57,209)( 58,212)( 59,211)( 60,210)( 61,257)( 62,260)( 63,259)
( 64,258)( 65,253)( 66,256)( 67,255)( 68,254)( 69,249)( 70,252)( 71,251)
( 72,250)( 73,245)( 74,248)( 75,247)( 76,246)( 77,241)( 78,244)( 79,243)
( 80,242)( 81,237)( 82,240)( 83,239)( 84,238)( 85,233)( 86,236)( 87,235)
( 88,234)( 89,229)( 90,232)( 91,231)( 92,230)( 93,225)( 94,228)( 95,227)
( 96,226)( 97,221)( 98,224)( 99,223)(100,222)(101,217)(102,220)(103,219)
(104,218)(105,161)(106,164)(107,163)(108,162)(109,157)(110,160)(111,159)
(112,158)(113,205)(114,208)(115,207)(116,206)(117,201)(118,204)(119,203)
(120,202)(121,197)(122,200)(123,199)(124,198)(125,193)(126,196)(127,195)
(128,194)(129,189)(130,192)(131,191)(132,190)(133,185)(134,188)(135,187)
(136,186)(137,181)(138,184)(139,183)(140,182)(141,177)(142,180)(143,179)
(144,178)(145,173)(146,176)(147,175)(148,174)(149,169)(150,172)(151,171)
(152,170)(153,165)(154,168)(155,167)(156,166);
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*s2*s1*s0*s1*s2*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*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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