Polytope of Type {78,4}

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