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