# Polytope of Type {4,78}

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