Polytope of Type {4,237}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,237}*1896
if this polytope has a name.
Group : SmallGroup(1896,43)
Rank : 3
Schlafli Type : {4,237}
Number of vertices, edges, etc : 4, 474, 237
Order of s₀s₁s₂ : 237
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   79-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)(313,315)(314,316);;
s1 := (  3,  4)(  5,313)(  6,314)(  7,316)(  8,315)(  9,309)( 10,310)( 11,312)
( 12,311)( 13,305)( 14,306)( 15,308)( 16,307)( 17,301)( 18,302)( 19,304)
( 20,303)( 21,297)( 22,298)( 23,300)( 24,299)( 25,293)( 26,294)( 27,296)
( 28,295)( 29,289)( 30,290)( 31,292)( 32,291)( 33,285)( 34,286)( 35,288)
( 36,287)( 37,281)( 38,282)( 39,284)( 40,283)( 41,277)( 42,278)( 43,280)
( 44,279)( 45,273)( 46,274)( 47,276)( 48,275)( 49,269)( 50,270)( 51,272)
( 52,271)( 53,265)( 54,266)( 55,268)( 56,267)( 57,261)( 58,262)( 59,264)
( 60,263)( 61,257)( 62,258)( 63,260)( 64,259)( 65,253)( 66,254)( 67,256)
( 68,255)( 69,249)( 70,250)( 71,252)( 72,251)( 73,245)( 74,246)( 75,248)
( 76,247)( 77,241)( 78,242)( 79,244)( 80,243)( 81,237)( 82,238)( 83,240)
( 84,239)( 85,233)( 86,234)( 87,236)( 88,235)( 89,229)( 90,230)( 91,232)
( 92,231)( 93,225)( 94,226)( 95,228)( 96,227)( 97,221)( 98,222)( 99,224)
(100,223)(101,217)(102,218)(103,220)(104,219)(105,213)(106,214)(107,216)
(108,215)(109,209)(110,210)(111,212)(112,211)(113,205)(114,206)(115,208)
(116,207)(117,201)(118,202)(119,204)(120,203)(121,197)(122,198)(123,200)
(124,199)(125,193)(126,194)(127,196)(128,195)(129,189)(130,190)(131,192)
(132,191)(133,185)(134,186)(135,188)(136,187)(137,181)(138,182)(139,184)
(140,183)(141,177)(142,178)(143,180)(144,179)(145,173)(146,174)(147,176)
(148,175)(149,169)(150,170)(151,172)(152,171)(153,165)(154,166)(155,168)
(156,167)(157,161)(158,162)(159,164)(160,163);;
s2 := (  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9,313)( 10,316)( 11,315)( 12,314)
( 13,309)( 14,312)( 15,311)( 16,310)( 17,305)( 18,308)( 19,307)( 20,306)
( 21,301)( 22,304)( 23,303)( 24,302)( 25,297)( 26,300)( 27,299)( 28,298)
( 29,293)( 30,296)( 31,295)( 32,294)( 33,289)( 34,292)( 35,291)( 36,290)
( 37,285)( 38,288)( 39,287)( 40,286)( 41,281)( 42,284)( 43,283)( 44,282)
( 45,277)( 46,280)( 47,279)( 48,278)( 49,273)( 50,276)( 51,275)( 52,274)
( 53,269)( 54,272)( 55,271)( 56,270)( 57,265)( 58,268)( 59,267)( 60,266)
( 61,261)( 62,264)( 63,263)( 64,262)( 65,257)( 66,260)( 67,259)( 68,258)
( 69,253)( 70,256)( 71,255)( 72,254)( 73,249)( 74,252)( 75,251)( 76,250)
( 77,245)( 78,248)( 79,247)( 80,246)( 81,241)( 82,244)( 83,243)( 84,242)
( 85,237)( 86,240)( 87,239)( 88,238)( 89,233)( 90,236)( 91,235)( 92,234)
( 93,229)( 94,232)( 95,231)( 96,230)( 97,225)( 98,228)( 99,227)(100,226)
(101,221)(102,224)(103,223)(104,222)(105,217)(106,220)(107,219)(108,218)
(109,213)(110,216)(111,215)(112,214)(113,209)(114,212)(115,211)(116,210)
(117,205)(118,208)(119,207)(120,206)(121,201)(122,204)(123,203)(124,202)
(125,197)(126,200)(127,199)(128,198)(129,193)(130,196)(131,195)(132,194)
(133,189)(134,192)(135,191)(136,190)(137,185)(138,188)(139,187)(140,186)
(141,181)(142,184)(143,183)(144,182)(145,177)(146,180)(147,179)(148,178)
(149,173)(150,176)(151,175)(152,174)(153,169)(154,172)(155,171)(156,170)
(157,165)(158,168)(159,167)(160,166)(162,164);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(316)!(  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)(313,315)(314,316);
s1 := Sym(316)!(  3,  4)(  5,313)(  6,314)(  7,316)(  8,315)(  9,309)( 10,310)
( 11,312)( 12,311)( 13,305)( 14,306)( 15,308)( 16,307)( 17,301)( 18,302)
( 19,304)( 20,303)( 21,297)( 22,298)( 23,300)( 24,299)( 25,293)( 26,294)
( 27,296)( 28,295)( 29,289)( 30,290)( 31,292)( 32,291)( 33,285)( 34,286)
( 35,288)( 36,287)( 37,281)( 38,282)( 39,284)( 40,283)( 41,277)( 42,278)
( 43,280)( 44,279)( 45,273)( 46,274)( 47,276)( 48,275)( 49,269)( 50,270)
( 51,272)( 52,271)( 53,265)( 54,266)( 55,268)( 56,267)( 57,261)( 58,262)
( 59,264)( 60,263)( 61,257)( 62,258)( 63,260)( 64,259)( 65,253)( 66,254)
( 67,256)( 68,255)( 69,249)( 70,250)( 71,252)( 72,251)( 73,245)( 74,246)
( 75,248)( 76,247)( 77,241)( 78,242)( 79,244)( 80,243)( 81,237)( 82,238)
( 83,240)( 84,239)( 85,233)( 86,234)( 87,236)( 88,235)( 89,229)( 90,230)
( 91,232)( 92,231)( 93,225)( 94,226)( 95,228)( 96,227)( 97,221)( 98,222)
( 99,224)(100,223)(101,217)(102,218)(103,220)(104,219)(105,213)(106,214)
(107,216)(108,215)(109,209)(110,210)(111,212)(112,211)(113,205)(114,206)
(115,208)(116,207)(117,201)(118,202)(119,204)(120,203)(121,197)(122,198)
(123,200)(124,199)(125,193)(126,194)(127,196)(128,195)(129,189)(130,190)
(131,192)(132,191)(133,185)(134,186)(135,188)(136,187)(137,181)(138,182)
(139,184)(140,183)(141,177)(142,178)(143,180)(144,179)(145,173)(146,174)
(147,176)(148,175)(149,169)(150,170)(151,172)(152,171)(153,165)(154,166)
(155,168)(156,167)(157,161)(158,162)(159,164)(160,163);
s2 := Sym(316)!(  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9,313)( 10,316)( 11,315)
( 12,314)( 13,309)( 14,312)( 15,311)( 16,310)( 17,305)( 18,308)( 19,307)
( 20,306)( 21,301)( 22,304)( 23,303)( 24,302)( 25,297)( 26,300)( 27,299)
( 28,298)( 29,293)( 30,296)( 31,295)( 32,294)( 33,289)( 34,292)( 35,291)
( 36,290)( 37,285)( 38,288)( 39,287)( 40,286)( 41,281)( 42,284)( 43,283)
( 44,282)( 45,277)( 46,280)( 47,279)( 48,278)( 49,273)( 50,276)( 51,275)
( 52,274)( 53,269)( 54,272)( 55,271)( 56,270)( 57,265)( 58,268)( 59,267)
( 60,266)( 61,261)( 62,264)( 63,263)( 64,262)( 65,257)( 66,260)( 67,259)
( 68,258)( 69,253)( 70,256)( 71,255)( 72,254)( 73,249)( 74,252)( 75,251)
( 76,250)( 77,245)( 78,248)( 79,247)( 80,246)( 81,241)( 82,244)( 83,243)
( 84,242)( 85,237)( 86,240)( 87,239)( 88,238)( 89,233)( 90,236)( 91,235)
( 92,234)( 93,229)( 94,232)( 95,231)( 96,230)( 97,225)( 98,228)( 99,227)
(100,226)(101,221)(102,224)(103,223)(104,222)(105,217)(106,220)(107,219)
(108,218)(109,213)(110,216)(111,215)(112,214)(113,209)(114,212)(115,211)
(116,210)(117,205)(118,208)(119,207)(120,206)(121,201)(122,204)(123,203)
(124,202)(125,197)(126,200)(127,199)(128,198)(129,193)(130,196)(131,195)
(132,194)(133,189)(134,192)(135,191)(136,190)(137,185)(138,188)(139,187)
(140,186)(141,181)(142,184)(143,183)(144,182)(145,177)(146,180)(147,179)
(148,178)(149,173)(150,176)(151,175)(152,174)(153,169)(154,172)(155,171)
(156,170)(157,165)(158,168)(159,167)(160,166)(162,164);
poly := sub<Sym(316)|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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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