Polytope of Type {2,4,39}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,39}*1248
if this polytope has a name.
Group : SmallGroup(1248,1441)
Rank : 4
Schlafli Type : {2,4,39}
Number of vertices, edges, etc : 2, 8, 156, 78
Order of s0s1s2s3 : 78
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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
4-fold quotients : {2,2,39}*312
12-fold quotients : {2,2,13}*104
13-fold quotients : {2,4,3}*96
26-fold quotients : {2,4,3}*48
52-fold quotients : {2,2,3}*24
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 := ( 3,111)( 4,114)( 5,113)( 6,112)( 7,107)( 8,110)( 9,109)( 10,108)
( 11,155)( 12,158)( 13,157)( 14,156)( 15,151)( 16,154)( 17,153)( 18,152)
( 19,147)( 20,150)( 21,149)( 22,148)( 23,143)( 24,146)( 25,145)( 26,144)
( 27,139)( 28,142)( 29,141)( 30,140)( 31,135)( 32,138)( 33,137)( 34,136)
( 35,131)( 36,134)( 37,133)( 38,132)( 39,127)( 40,130)( 41,129)( 42,128)
( 43,123)( 44,126)( 45,125)( 46,124)( 47,119)( 48,122)( 49,121)( 50,120)
( 51,115)( 52,118)( 53,117)( 54,116)( 55, 59)( 56, 62)( 57, 61)( 58, 60)
( 63,103)( 64,106)( 65,105)( 66,104)( 67, 99)( 68,102)( 69,101)( 70,100)
( 71, 95)( 72, 98)( 73, 97)( 74, 96)( 75, 91)( 76, 94)( 77, 93)( 78, 92)
( 79, 87)( 80, 90)( 81, 89)( 82, 88)( 84, 86)(159,267)(160,270)(161,269)
(162,268)(163,263)(164,266)(165,265)(166,264)(167,311)(168,314)(169,313)
(170,312)(171,307)(172,310)(173,309)(174,308)(175,303)(176,306)(177,305)
(178,304)(179,299)(180,302)(181,301)(182,300)(183,295)(184,298)(185,297)
(186,296)(187,291)(188,294)(189,293)(190,292)(191,287)(192,290)(193,289)
(194,288)(195,283)(196,286)(197,285)(198,284)(199,279)(200,282)(201,281)
(202,280)(203,275)(204,278)(205,277)(206,276)(207,271)(208,274)(209,273)
(210,272)(211,215)(212,218)(213,217)(214,216)(219,259)(220,262)(221,261)
(222,260)(223,255)(224,258)(225,257)(226,256)(227,251)(228,254)(229,253)
(230,252)(231,247)(232,250)(233,249)(234,248)(235,243)(236,246)(237,245)
(238,244)(240,242);;
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*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*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)!( 3,111)( 4,114)( 5,113)( 6,112)( 7,107)( 8,110)( 9,109)
( 10,108)( 11,155)( 12,158)( 13,157)( 14,156)( 15,151)( 16,154)( 17,153)
( 18,152)( 19,147)( 20,150)( 21,149)( 22,148)( 23,143)( 24,146)( 25,145)
( 26,144)( 27,139)( 28,142)( 29,141)( 30,140)( 31,135)( 32,138)( 33,137)
( 34,136)( 35,131)( 36,134)( 37,133)( 38,132)( 39,127)( 40,130)( 41,129)
( 42,128)( 43,123)( 44,126)( 45,125)( 46,124)( 47,119)( 48,122)( 49,121)
( 50,120)( 51,115)( 52,118)( 53,117)( 54,116)( 55, 59)( 56, 62)( 57, 61)
( 58, 60)( 63,103)( 64,106)( 65,105)( 66,104)( 67, 99)( 68,102)( 69,101)
( 70,100)( 71, 95)( 72, 98)( 73, 97)( 74, 96)( 75, 91)( 76, 94)( 77, 93)
( 78, 92)( 79, 87)( 80, 90)( 81, 89)( 82, 88)( 84, 86)(159,267)(160,270)
(161,269)(162,268)(163,263)(164,266)(165,265)(166,264)(167,311)(168,314)
(169,313)(170,312)(171,307)(172,310)(173,309)(174,308)(175,303)(176,306)
(177,305)(178,304)(179,299)(180,302)(181,301)(182,300)(183,295)(184,298)
(185,297)(186,296)(187,291)(188,294)(189,293)(190,292)(191,287)(192,290)
(193,289)(194,288)(195,283)(196,286)(197,285)(198,284)(199,279)(200,282)
(201,281)(202,280)(203,275)(204,278)(205,277)(206,276)(207,271)(208,274)
(209,273)(210,272)(211,215)(212,218)(213,217)(214,216)(219,259)(220,262)
(221,261)(222,260)(223,255)(224,258)(225,257)(226,256)(227,251)(228,254)
(229,253)(230,252)(231,247)(232,250)(233,249)(234,248)(235,243)(236,246)
(237,245)(238,244)(240,242);
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*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*s2*s3*s2*s3 >;
to this polytope