Polytope of Type {4,57,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,57,4}*1824
if this polytope has a name.
Group : SmallGroup(1824,1249)
Rank : 4
Schlafli Type : {4,57,4}
Number of vertices, edges, etc : 4, 114, 114, 4
Order of s0s1s2s3 : 57
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
19-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)( 8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144)
(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)(152,160)
(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)(168,176)
(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)(184,192)
(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)(200,208)
(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)(216,224)
(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)(232,240)
(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)(248,256)
(257,265)(258,266)(259,267)(260,268)(261,269)(262,270)(263,271)(264,272)
(273,281)(274,282)(275,283)(276,284)(277,285)(278,286)(279,287)(280,288)
(289,297)(290,298)(291,299)(292,300)(293,301)(294,302)(295,303)(296,304);;
s1 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17,289)( 18,290)
( 19,292)( 20,291)( 21,293)( 22,294)( 23,296)( 24,295)( 25,301)( 26,302)
( 27,304)( 28,303)( 29,297)( 30,298)( 31,300)( 32,299)( 33,273)( 34,274)
( 35,276)( 36,275)( 37,277)( 38,278)( 39,280)( 40,279)( 41,285)( 42,286)
( 43,288)( 44,287)( 45,281)( 46,282)( 47,284)( 48,283)( 49,257)( 50,258)
( 51,260)( 52,259)( 53,261)( 54,262)( 55,264)( 56,263)( 57,269)( 58,270)
( 59,272)( 60,271)( 61,265)( 62,266)( 63,268)( 64,267)( 65,241)( 66,242)
( 67,244)( 68,243)( 69,245)( 70,246)( 71,248)( 72,247)( 73,253)( 74,254)
( 75,256)( 76,255)( 77,249)( 78,250)( 79,252)( 80,251)( 81,225)( 82,226)
( 83,228)( 84,227)( 85,229)( 86,230)( 87,232)( 88,231)( 89,237)( 90,238)
( 91,240)( 92,239)( 93,233)( 94,234)( 95,236)( 96,235)( 97,209)( 98,210)
( 99,212)(100,211)(101,213)(102,214)(103,216)(104,215)(105,221)(106,222)
(107,224)(108,223)(109,217)(110,218)(111,220)(112,219)(113,193)(114,194)
(115,196)(116,195)(117,197)(118,198)(119,200)(120,199)(121,205)(122,206)
(123,208)(124,207)(125,201)(126,202)(127,204)(128,203)(129,177)(130,178)
(131,180)(132,179)(133,181)(134,182)(135,184)(136,183)(137,189)(138,190)
(139,192)(140,191)(141,185)(142,186)(143,188)(144,187)(145,161)(146,162)
(147,164)(148,163)(149,165)(150,166)(151,168)(152,167)(153,173)(154,174)
(155,176)(156,175)(157,169)(158,170)(159,172)(160,171);;
s2 := ( 1, 17)( 2, 20)( 3, 19)( 4, 18)( 5, 29)( 6, 32)( 7, 31)( 8, 30)
( 9, 25)( 10, 28)( 11, 27)( 12, 26)( 13, 21)( 14, 24)( 15, 23)( 16, 22)
( 33,289)( 34,292)( 35,291)( 36,290)( 37,301)( 38,304)( 39,303)( 40,302)
( 41,297)( 42,300)( 43,299)( 44,298)( 45,293)( 46,296)( 47,295)( 48,294)
( 49,273)( 50,276)( 51,275)( 52,274)( 53,285)( 54,288)( 55,287)( 56,286)
( 57,281)( 58,284)( 59,283)( 60,282)( 61,277)( 62,280)( 63,279)( 64,278)
( 65,257)( 66,260)( 67,259)( 68,258)( 69,269)( 70,272)( 71,271)( 72,270)
( 73,265)( 74,268)( 75,267)( 76,266)( 77,261)( 78,264)( 79,263)( 80,262)
( 81,241)( 82,244)( 83,243)( 84,242)( 85,253)( 86,256)( 87,255)( 88,254)
( 89,249)( 90,252)( 91,251)( 92,250)( 93,245)( 94,248)( 95,247)( 96,246)
( 97,225)( 98,228)( 99,227)(100,226)(101,237)(102,240)(103,239)(104,238)
(105,233)(106,236)(107,235)(108,234)(109,229)(110,232)(111,231)(112,230)
(113,209)(114,212)(115,211)(116,210)(117,221)(118,224)(119,223)(120,222)
(121,217)(122,220)(123,219)(124,218)(125,213)(126,216)(127,215)(128,214)
(129,193)(130,196)(131,195)(132,194)(133,205)(134,208)(135,207)(136,206)
(137,201)(138,204)(139,203)(140,202)(141,197)(142,200)(143,199)(144,198)
(145,177)(146,180)(147,179)(148,178)(149,189)(150,192)(151,191)(152,190)
(153,185)(154,188)(155,187)(156,186)(157,181)(158,184)(159,183)(160,182)
(162,164)(165,173)(166,176)(167,175)(168,174)(170,172);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s3*s2*s1*s3*s2*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*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(304)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 13)( 6, 14)( 7, 15)
( 8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144)(145,153)(146,154)(147,155)(148,156)(149,157)(150,158)(151,159)
(152,160)(161,169)(162,170)(163,171)(164,172)(165,173)(166,174)(167,175)
(168,176)(177,185)(178,186)(179,187)(180,188)(181,189)(182,190)(183,191)
(184,192)(193,201)(194,202)(195,203)(196,204)(197,205)(198,206)(199,207)
(200,208)(209,217)(210,218)(211,219)(212,220)(213,221)(214,222)(215,223)
(216,224)(225,233)(226,234)(227,235)(228,236)(229,237)(230,238)(231,239)
(232,240)(241,249)(242,250)(243,251)(244,252)(245,253)(246,254)(247,255)
(248,256)(257,265)(258,266)(259,267)(260,268)(261,269)(262,270)(263,271)
(264,272)(273,281)(274,282)(275,283)(276,284)(277,285)(278,286)(279,287)
(280,288)(289,297)(290,298)(291,299)(292,300)(293,301)(294,302)(295,303)
(296,304);
s1 := Sym(304)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17,289)
( 18,290)( 19,292)( 20,291)( 21,293)( 22,294)( 23,296)( 24,295)( 25,301)
( 26,302)( 27,304)( 28,303)( 29,297)( 30,298)( 31,300)( 32,299)( 33,273)
( 34,274)( 35,276)( 36,275)( 37,277)( 38,278)( 39,280)( 40,279)( 41,285)
( 42,286)( 43,288)( 44,287)( 45,281)( 46,282)( 47,284)( 48,283)( 49,257)
( 50,258)( 51,260)( 52,259)( 53,261)( 54,262)( 55,264)( 56,263)( 57,269)
( 58,270)( 59,272)( 60,271)( 61,265)( 62,266)( 63,268)( 64,267)( 65,241)
( 66,242)( 67,244)( 68,243)( 69,245)( 70,246)( 71,248)( 72,247)( 73,253)
( 74,254)( 75,256)( 76,255)( 77,249)( 78,250)( 79,252)( 80,251)( 81,225)
( 82,226)( 83,228)( 84,227)( 85,229)( 86,230)( 87,232)( 88,231)( 89,237)
( 90,238)( 91,240)( 92,239)( 93,233)( 94,234)( 95,236)( 96,235)( 97,209)
( 98,210)( 99,212)(100,211)(101,213)(102,214)(103,216)(104,215)(105,221)
(106,222)(107,224)(108,223)(109,217)(110,218)(111,220)(112,219)(113,193)
(114,194)(115,196)(116,195)(117,197)(118,198)(119,200)(120,199)(121,205)
(122,206)(123,208)(124,207)(125,201)(126,202)(127,204)(128,203)(129,177)
(130,178)(131,180)(132,179)(133,181)(134,182)(135,184)(136,183)(137,189)
(138,190)(139,192)(140,191)(141,185)(142,186)(143,188)(144,187)(145,161)
(146,162)(147,164)(148,163)(149,165)(150,166)(151,168)(152,167)(153,173)
(154,174)(155,176)(156,175)(157,169)(158,170)(159,172)(160,171);
s2 := Sym(304)!( 1, 17)( 2, 20)( 3, 19)( 4, 18)( 5, 29)( 6, 32)( 7, 31)
( 8, 30)( 9, 25)( 10, 28)( 11, 27)( 12, 26)( 13, 21)( 14, 24)( 15, 23)
( 16, 22)( 33,289)( 34,292)( 35,291)( 36,290)( 37,301)( 38,304)( 39,303)
( 40,302)( 41,297)( 42,300)( 43,299)( 44,298)( 45,293)( 46,296)( 47,295)
( 48,294)( 49,273)( 50,276)( 51,275)( 52,274)( 53,285)( 54,288)( 55,287)
( 56,286)( 57,281)( 58,284)( 59,283)( 60,282)( 61,277)( 62,280)( 63,279)
( 64,278)( 65,257)( 66,260)( 67,259)( 68,258)( 69,269)( 70,272)( 71,271)
( 72,270)( 73,265)( 74,268)( 75,267)( 76,266)( 77,261)( 78,264)( 79,263)
( 80,262)( 81,241)( 82,244)( 83,243)( 84,242)( 85,253)( 86,256)( 87,255)
( 88,254)( 89,249)( 90,252)( 91,251)( 92,250)( 93,245)( 94,248)( 95,247)
( 96,246)( 97,225)( 98,228)( 99,227)(100,226)(101,237)(102,240)(103,239)
(104,238)(105,233)(106,236)(107,235)(108,234)(109,229)(110,232)(111,231)
(112,230)(113,209)(114,212)(115,211)(116,210)(117,221)(118,224)(119,223)
(120,222)(121,217)(122,220)(123,219)(124,218)(125,213)(126,216)(127,215)
(128,214)(129,193)(130,196)(131,195)(132,194)(133,205)(134,208)(135,207)
(136,206)(137,201)(138,204)(139,203)(140,202)(141,197)(142,200)(143,199)
(144,198)(145,177)(146,180)(147,179)(148,178)(149,189)(150,192)(151,191)
(152,190)(153,185)(154,188)(155,187)(156,186)(157,181)(158,184)(159,183)
(160,182)(162,164)(165,173)(166,176)(167,175)(168,174)(170,172);
s3 := Sym(304)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304);
poly := sub<Sym(304)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s3*s2*s1*s3*s2*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*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