Polytope of Type {2,78,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,78,4}*1248c
if this polytope has a name.
Group : SmallGroup(1248,1441)
Rank : 4
Schlafli Type : {2,78,4}
Number of vertices, edges, etc : 2, 78, 156, 4
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 : {2,39,4}*624
13-fold quotients : {2,6,4}*96b
26-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 3,215)( 4,216)( 5,218)( 6,217)( 7,211)( 8,212)( 9,214)( 10,213)
( 11,259)( 12,260)( 13,262)( 14,261)( 15,255)( 16,256)( 17,258)( 18,257)
( 19,251)( 20,252)( 21,254)( 22,253)( 23,247)( 24,248)( 25,250)( 26,249)
( 27,243)( 28,244)( 29,246)( 30,245)( 31,239)( 32,240)( 33,242)( 34,241)
( 35,235)( 36,236)( 37,238)( 38,237)( 39,231)( 40,232)( 41,234)( 42,233)
( 43,227)( 44,228)( 45,230)( 46,229)( 47,223)( 48,224)( 49,226)( 50,225)
( 51,219)( 52,220)( 53,222)( 54,221)( 55,163)( 56,164)( 57,166)( 58,165)
( 59,159)( 60,160)( 61,162)( 62,161)( 63,207)( 64,208)( 65,210)( 66,209)
( 67,203)( 68,204)( 69,206)( 70,205)( 71,199)( 72,200)( 73,202)( 74,201)
( 75,195)( 76,196)( 77,198)( 78,197)( 79,191)( 80,192)( 81,194)( 82,193)
( 83,187)( 84,188)( 85,190)( 86,189)( 87,183)( 88,184)( 89,186)( 90,185)
( 91,179)( 92,180)( 93,182)( 94,181)( 95,175)( 96,176)( 97,178)( 98,177)
( 99,171)(100,172)(101,174)(102,173)(103,167)(104,168)(105,170)(106,169)
(107,267)(108,268)(109,270)(110,269)(111,263)(112,264)(113,266)(114,265)
(115,311)(116,312)(117,314)(118,313)(119,307)(120,308)(121,310)(122,309)
(123,303)(124,304)(125,306)(126,305)(127,299)(128,300)(129,302)(130,301)
(131,295)(132,296)(133,298)(134,297)(135,291)(136,292)(137,294)(138,293)
(139,287)(140,288)(141,290)(142,289)(143,283)(144,284)(145,286)(146,285)
(147,279)(148,280)(149,282)(150,281)(151,275)(152,276)(153,278)(154,277)
(155,271)(156,272)(157,274)(158,273);;
s3 := ( 3,162)( 4,161)( 5,160)( 6,159)( 7,166)( 8,165)( 9,164)( 10,163)
( 11,170)( 12,169)( 13,168)( 14,167)( 15,174)( 16,173)( 17,172)( 18,171)
( 19,178)( 20,177)( 21,176)( 22,175)( 23,182)( 24,181)( 25,180)( 26,179)
( 27,186)( 28,185)( 29,184)( 30,183)( 31,190)( 32,189)( 33,188)( 34,187)
( 35,194)( 36,193)( 37,192)( 38,191)( 39,198)( 40,197)( 41,196)( 42,195)
( 43,202)( 44,201)( 45,200)( 46,199)( 47,206)( 48,205)( 49,204)( 50,203)
( 51,210)( 52,209)( 53,208)( 54,207)( 55,214)( 56,213)( 57,212)( 58,211)
( 59,218)( 60,217)( 61,216)( 62,215)( 63,222)( 64,221)( 65,220)( 66,219)
( 67,226)( 68,225)( 69,224)( 70,223)( 71,230)( 72,229)( 73,228)( 74,227)
( 75,234)( 76,233)( 77,232)( 78,231)( 79,238)( 80,237)( 81,236)( 82,235)
( 83,242)( 84,241)( 85,240)( 86,239)( 87,246)( 88,245)( 89,244)( 90,243)
( 91,250)( 92,249)( 93,248)( 94,247)( 95,254)( 96,253)( 97,252)( 98,251)
( 99,258)(100,257)(101,256)(102,255)(103,262)(104,261)(105,260)(106,259)
(107,266)(108,265)(109,264)(110,263)(111,270)(112,269)(113,268)(114,267)
(115,274)(116,273)(117,272)(118,271)(119,278)(120,277)(121,276)(122,275)
(123,282)(124,281)(125,280)(126,279)(127,286)(128,285)(129,284)(130,283)
(131,290)(132,289)(133,288)(134,287)(135,294)(136,293)(137,292)(138,291)
(139,298)(140,297)(141,296)(142,295)(143,302)(144,301)(145,300)(146,299)
(147,306)(148,305)(149,304)(150,303)(151,310)(152,309)(153,308)(154,307)
(155,314)(156,313)(157,312)(158,311);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*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(314)!(1,2);
s1 := 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);
s2 := Sym(314)!( 3,215)( 4,216)( 5,218)( 6,217)( 7,211)( 8,212)( 9,214)
( 10,213)( 11,259)( 12,260)( 13,262)( 14,261)( 15,255)( 16,256)( 17,258)
( 18,257)( 19,251)( 20,252)( 21,254)( 22,253)( 23,247)( 24,248)( 25,250)
( 26,249)( 27,243)( 28,244)( 29,246)( 30,245)( 31,239)( 32,240)( 33,242)
( 34,241)( 35,235)( 36,236)( 37,238)( 38,237)( 39,231)( 40,232)( 41,234)
( 42,233)( 43,227)( 44,228)( 45,230)( 46,229)( 47,223)( 48,224)( 49,226)
( 50,225)( 51,219)( 52,220)( 53,222)( 54,221)( 55,163)( 56,164)( 57,166)
( 58,165)( 59,159)( 60,160)( 61,162)( 62,161)( 63,207)( 64,208)( 65,210)
( 66,209)( 67,203)( 68,204)( 69,206)( 70,205)( 71,199)( 72,200)( 73,202)
( 74,201)( 75,195)( 76,196)( 77,198)( 78,197)( 79,191)( 80,192)( 81,194)
( 82,193)( 83,187)( 84,188)( 85,190)( 86,189)( 87,183)( 88,184)( 89,186)
( 90,185)( 91,179)( 92,180)( 93,182)( 94,181)( 95,175)( 96,176)( 97,178)
( 98,177)( 99,171)(100,172)(101,174)(102,173)(103,167)(104,168)(105,170)
(106,169)(107,267)(108,268)(109,270)(110,269)(111,263)(112,264)(113,266)
(114,265)(115,311)(116,312)(117,314)(118,313)(119,307)(120,308)(121,310)
(122,309)(123,303)(124,304)(125,306)(126,305)(127,299)(128,300)(129,302)
(130,301)(131,295)(132,296)(133,298)(134,297)(135,291)(136,292)(137,294)
(138,293)(139,287)(140,288)(141,290)(142,289)(143,283)(144,284)(145,286)
(146,285)(147,279)(148,280)(149,282)(150,281)(151,275)(152,276)(153,278)
(154,277)(155,271)(156,272)(157,274)(158,273);
s3 := Sym(314)!( 3,162)( 4,161)( 5,160)( 6,159)( 7,166)( 8,165)( 9,164)
( 10,163)( 11,170)( 12,169)( 13,168)( 14,167)( 15,174)( 16,173)( 17,172)
( 18,171)( 19,178)( 20,177)( 21,176)( 22,175)( 23,182)( 24,181)( 25,180)
( 26,179)( 27,186)( 28,185)( 29,184)( 30,183)( 31,190)( 32,189)( 33,188)
( 34,187)( 35,194)( 36,193)( 37,192)( 38,191)( 39,198)( 40,197)( 41,196)
( 42,195)( 43,202)( 44,201)( 45,200)( 46,199)( 47,206)( 48,205)( 49,204)
( 50,203)( 51,210)( 52,209)( 53,208)( 54,207)( 55,214)( 56,213)( 57,212)
( 58,211)( 59,218)( 60,217)( 61,216)( 62,215)( 63,222)( 64,221)( 65,220)
( 66,219)( 67,226)( 68,225)( 69,224)( 70,223)( 71,230)( 72,229)( 73,228)
( 74,227)( 75,234)( 76,233)( 77,232)( 78,231)( 79,238)( 80,237)( 81,236)
( 82,235)( 83,242)( 84,241)( 85,240)( 86,239)( 87,246)( 88,245)( 89,244)
( 90,243)( 91,250)( 92,249)( 93,248)( 94,247)( 95,254)( 96,253)( 97,252)
( 98,251)( 99,258)(100,257)(101,256)(102,255)(103,262)(104,261)(105,260)
(106,259)(107,266)(108,265)(109,264)(110,263)(111,270)(112,269)(113,268)
(114,267)(115,274)(116,273)(117,272)(118,271)(119,278)(120,277)(121,276)
(122,275)(123,282)(124,281)(125,280)(126,279)(127,286)(128,285)(129,284)
(130,283)(131,290)(132,289)(133,288)(134,287)(135,294)(136,293)(137,292)
(138,291)(139,298)(140,297)(141,296)(142,295)(143,302)(144,301)(145,300)
(146,299)(147,306)(148,305)(149,304)(150,303)(151,310)(152,309)(153,308)
(154,307)(155,314)(156,313)(157,312)(158,311);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*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 >;
to this polytope