Polytope of Type {2,4,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,78}*1248c
if this polytope has a name.
Group : SmallGroup(1248,1441)
Rank : 4
Schlafli Type : {2,4,78}
Number of vertices, edges, etc : 2, 4, 156, 78
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,4,39}*624
   13-fold quotients : {2,4,6}*96b
   26-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3,161)(  4,162)(  5,159)(  6,160)(  7,165)(  8,166)(  9,163)( 10,164)
( 11,169)( 12,170)( 13,167)( 14,168)( 15,173)( 16,174)( 17,171)( 18,172)
( 19,177)( 20,178)( 21,175)( 22,176)( 23,181)( 24,182)( 25,179)( 26,180)
( 27,185)( 28,186)( 29,183)( 30,184)( 31,189)( 32,190)( 33,187)( 34,188)
( 35,193)( 36,194)( 37,191)( 38,192)( 39,197)( 40,198)( 41,195)( 42,196)
( 43,201)( 44,202)( 45,199)( 46,200)( 47,205)( 48,206)( 49,203)( 50,204)
( 51,209)( 52,210)( 53,207)( 54,208)( 55,213)( 56,214)( 57,211)( 58,212)
( 59,217)( 60,218)( 61,215)( 62,216)( 63,221)( 64,222)( 65,219)( 66,220)
( 67,225)( 68,226)( 69,223)( 70,224)( 71,229)( 72,230)( 73,227)( 74,228)
( 75,233)( 76,234)( 77,231)( 78,232)( 79,237)( 80,238)( 81,235)( 82,236)
( 83,241)( 84,242)( 85,239)( 86,240)( 87,245)( 88,246)( 89,243)( 90,244)
( 91,249)( 92,250)( 93,247)( 94,248)( 95,253)( 96,254)( 97,251)( 98,252)
( 99,257)(100,258)(101,255)(102,256)(103,261)(104,262)(105,259)(106,260)
(107,265)(108,266)(109,263)(110,264)(111,269)(112,270)(113,267)(114,268)
(115,273)(116,274)(117,271)(118,272)(119,277)(120,278)(121,275)(122,276)
(123,281)(124,282)(125,279)(126,280)(127,285)(128,286)(129,283)(130,284)
(131,289)(132,290)(133,287)(134,288)(135,293)(136,294)(137,291)(138,292)
(139,297)(140,298)(141,295)(142,296)(143,301)(144,302)(145,299)(146,300)
(147,305)(148,306)(149,303)(150,304)(151,309)(152,310)(153,307)(154,308)
(155,313)(156,314)(157,311)(158,312);;
s2 := (  4,  5)(  7, 51)(  8, 53)(  9, 52)( 10, 54)( 11, 47)( 12, 49)( 13, 48)
( 14, 50)( 15, 43)( 16, 45)( 17, 44)( 18, 46)( 19, 39)( 20, 41)( 21, 40)
( 22, 42)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 27, 31)( 28, 33)( 29, 32)
( 30, 34)( 55,107)( 56,109)( 57,108)( 58,110)( 59,155)( 60,157)( 61,156)
( 62,158)( 63,151)( 64,153)( 65,152)( 66,154)( 67,147)( 68,149)( 69,148)
( 70,150)( 71,143)( 72,145)( 73,144)( 74,146)( 75,139)( 76,141)( 77,140)
( 78,142)( 79,135)( 80,137)( 81,136)( 82,138)( 83,131)( 84,133)( 85,132)
( 86,134)( 87,127)( 88,129)( 89,128)( 90,130)( 91,123)( 92,125)( 93,124)
( 94,126)( 95,119)( 96,121)( 97,120)( 98,122)( 99,115)(100,117)(101,116)
(102,118)(103,111)(104,113)(105,112)(106,114)(160,161)(163,207)(164,209)
(165,208)(166,210)(167,203)(168,205)(169,204)(170,206)(171,199)(172,201)
(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,191)(180,193)
(181,192)(182,194)(183,187)(184,189)(185,188)(186,190)(211,263)(212,265)
(213,264)(214,266)(215,311)(216,313)(217,312)(218,314)(219,307)(220,309)
(221,308)(222,310)(223,303)(224,305)(225,304)(226,306)(227,299)(228,301)
(229,300)(230,302)(231,295)(232,297)(233,296)(234,298)(235,291)(236,293)
(237,292)(238,294)(239,287)(240,289)(241,288)(242,290)(243,283)(244,285)
(245,284)(246,286)(247,279)(248,281)(249,280)(250,282)(251,275)(252,277)
(253,276)(254,278)(255,271)(256,273)(257,272)(258,274)(259,267)(260,269)
(261,268)(262,270);;
s3 := (  3,267)(  4,270)(  5,269)(  6,268)(  7,263)(  8,266)(  9,265)( 10,264)
( 11,311)( 12,314)( 13,313)( 14,312)( 15,307)( 16,310)( 17,309)( 18,308)
( 19,303)( 20,306)( 21,305)( 22,304)( 23,299)( 24,302)( 25,301)( 26,300)
( 27,295)( 28,298)( 29,297)( 30,296)( 31,291)( 32,294)( 33,293)( 34,292)
( 35,287)( 36,290)( 37,289)( 38,288)( 39,283)( 40,286)( 41,285)( 42,284)
( 43,279)( 44,282)( 45,281)( 46,280)( 47,275)( 48,278)( 49,277)( 50,276)
( 51,271)( 52,274)( 53,273)( 54,272)( 55,215)( 56,218)( 57,217)( 58,216)
( 59,211)( 60,214)( 61,213)( 62,212)( 63,259)( 64,262)( 65,261)( 66,260)
( 67,255)( 68,258)( 69,257)( 70,256)( 71,251)( 72,254)( 73,253)( 74,252)
( 75,247)( 76,250)( 77,249)( 78,248)( 79,243)( 80,246)( 81,245)( 82,244)
( 83,239)( 84,242)( 85,241)( 86,240)( 87,235)( 88,238)( 89,237)( 90,236)
( 91,231)( 92,234)( 93,233)( 94,232)( 95,227)( 96,230)( 97,229)( 98,228)
( 99,223)(100,226)(101,225)(102,224)(103,219)(104,222)(105,221)(106,220)
(107,163)(108,166)(109,165)(110,164)(111,159)(112,162)(113,161)(114,160)
(115,207)(116,210)(117,209)(118,208)(119,203)(120,206)(121,205)(122,204)
(123,199)(124,202)(125,201)(126,200)(127,195)(128,198)(129,197)(130,196)
(131,191)(132,194)(133,193)(134,192)(135,187)(136,190)(137,189)(138,188)
(139,183)(140,186)(141,185)(142,184)(143,179)(144,182)(145,181)(146,180)
(147,175)(148,178)(149,177)(150,176)(151,171)(152,174)(153,173)(154,172)
(155,167)(156,170)(157,169)(158,168);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(314)!(1,2);
s1 := Sym(314)!(  3,161)(  4,162)(  5,159)(  6,160)(  7,165)(  8,166)(  9,163)
( 10,164)( 11,169)( 12,170)( 13,167)( 14,168)( 15,173)( 16,174)( 17,171)
( 18,172)( 19,177)( 20,178)( 21,175)( 22,176)( 23,181)( 24,182)( 25,179)
( 26,180)( 27,185)( 28,186)( 29,183)( 30,184)( 31,189)( 32,190)( 33,187)
( 34,188)( 35,193)( 36,194)( 37,191)( 38,192)( 39,197)( 40,198)( 41,195)
( 42,196)( 43,201)( 44,202)( 45,199)( 46,200)( 47,205)( 48,206)( 49,203)
( 50,204)( 51,209)( 52,210)( 53,207)( 54,208)( 55,213)( 56,214)( 57,211)
( 58,212)( 59,217)( 60,218)( 61,215)( 62,216)( 63,221)( 64,222)( 65,219)
( 66,220)( 67,225)( 68,226)( 69,223)( 70,224)( 71,229)( 72,230)( 73,227)
( 74,228)( 75,233)( 76,234)( 77,231)( 78,232)( 79,237)( 80,238)( 81,235)
( 82,236)( 83,241)( 84,242)( 85,239)( 86,240)( 87,245)( 88,246)( 89,243)
( 90,244)( 91,249)( 92,250)( 93,247)( 94,248)( 95,253)( 96,254)( 97,251)
( 98,252)( 99,257)(100,258)(101,255)(102,256)(103,261)(104,262)(105,259)
(106,260)(107,265)(108,266)(109,263)(110,264)(111,269)(112,270)(113,267)
(114,268)(115,273)(116,274)(117,271)(118,272)(119,277)(120,278)(121,275)
(122,276)(123,281)(124,282)(125,279)(126,280)(127,285)(128,286)(129,283)
(130,284)(131,289)(132,290)(133,287)(134,288)(135,293)(136,294)(137,291)
(138,292)(139,297)(140,298)(141,295)(142,296)(143,301)(144,302)(145,299)
(146,300)(147,305)(148,306)(149,303)(150,304)(151,309)(152,310)(153,307)
(154,308)(155,313)(156,314)(157,311)(158,312);
s2 := Sym(314)!(  4,  5)(  7, 51)(  8, 53)(  9, 52)( 10, 54)( 11, 47)( 12, 49)
( 13, 48)( 14, 50)( 15, 43)( 16, 45)( 17, 44)( 18, 46)( 19, 39)( 20, 41)
( 21, 40)( 22, 42)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 27, 31)( 28, 33)
( 29, 32)( 30, 34)( 55,107)( 56,109)( 57,108)( 58,110)( 59,155)( 60,157)
( 61,156)( 62,158)( 63,151)( 64,153)( 65,152)( 66,154)( 67,147)( 68,149)
( 69,148)( 70,150)( 71,143)( 72,145)( 73,144)( 74,146)( 75,139)( 76,141)
( 77,140)( 78,142)( 79,135)( 80,137)( 81,136)( 82,138)( 83,131)( 84,133)
( 85,132)( 86,134)( 87,127)( 88,129)( 89,128)( 90,130)( 91,123)( 92,125)
( 93,124)( 94,126)( 95,119)( 96,121)( 97,120)( 98,122)( 99,115)(100,117)
(101,116)(102,118)(103,111)(104,113)(105,112)(106,114)(160,161)(163,207)
(164,209)(165,208)(166,210)(167,203)(168,205)(169,204)(170,206)(171,199)
(172,201)(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,191)
(180,193)(181,192)(182,194)(183,187)(184,189)(185,188)(186,190)(211,263)
(212,265)(213,264)(214,266)(215,311)(216,313)(217,312)(218,314)(219,307)
(220,309)(221,308)(222,310)(223,303)(224,305)(225,304)(226,306)(227,299)
(228,301)(229,300)(230,302)(231,295)(232,297)(233,296)(234,298)(235,291)
(236,293)(237,292)(238,294)(239,287)(240,289)(241,288)(242,290)(243,283)
(244,285)(245,284)(246,286)(247,279)(248,281)(249,280)(250,282)(251,275)
(252,277)(253,276)(254,278)(255,271)(256,273)(257,272)(258,274)(259,267)
(260,269)(261,268)(262,270);
s3 := Sym(314)!(  3,267)(  4,270)(  5,269)(  6,268)(  7,263)(  8,266)(  9,265)
( 10,264)( 11,311)( 12,314)( 13,313)( 14,312)( 15,307)( 16,310)( 17,309)
( 18,308)( 19,303)( 20,306)( 21,305)( 22,304)( 23,299)( 24,302)( 25,301)
( 26,300)( 27,295)( 28,298)( 29,297)( 30,296)( 31,291)( 32,294)( 33,293)
( 34,292)( 35,287)( 36,290)( 37,289)( 38,288)( 39,283)( 40,286)( 41,285)
( 42,284)( 43,279)( 44,282)( 45,281)( 46,280)( 47,275)( 48,278)( 49,277)
( 50,276)( 51,271)( 52,274)( 53,273)( 54,272)( 55,215)( 56,218)( 57,217)
( 58,216)( 59,211)( 60,214)( 61,213)( 62,212)( 63,259)( 64,262)( 65,261)
( 66,260)( 67,255)( 68,258)( 69,257)( 70,256)( 71,251)( 72,254)( 73,253)
( 74,252)( 75,247)( 76,250)( 77,249)( 78,248)( 79,243)( 80,246)( 81,245)
( 82,244)( 83,239)( 84,242)( 85,241)( 86,240)( 87,235)( 88,238)( 89,237)
( 90,236)( 91,231)( 92,234)( 93,233)( 94,232)( 95,227)( 96,230)( 97,229)
( 98,228)( 99,223)(100,226)(101,225)(102,224)(103,219)(104,222)(105,221)
(106,220)(107,163)(108,166)(109,165)(110,164)(111,159)(112,162)(113,161)
(114,160)(115,207)(116,210)(117,209)(118,208)(119,203)(120,206)(121,205)
(122,204)(123,199)(124,202)(125,201)(126,200)(127,195)(128,198)(129,197)
(130,196)(131,191)(132,194)(133,193)(134,192)(135,187)(136,190)(137,189)
(138,188)(139,183)(140,186)(141,185)(142,184)(143,179)(144,182)(145,181)
(146,180)(147,175)(148,178)(149,177)(150,176)(151,171)(152,174)(153,173)
(154,172)(155,167)(156,170)(157,169)(158,168);
poly := sub<Sym(314)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope