Polytope of Type {4,231}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,231}*1848
if this polytope has a name.
Group : SmallGroup(1848,135)
Rank : 3
Schlafli Type : {4,231}
Number of vertices, edges, etc : 4, 462, 231
Order of s₀s₁s₂ : 231
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) :
   7-fold quotients : {4,33}*264
   11-fold quotients : {4,21}*168
   77-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);;
s1 := (  3,  4)(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 37)( 10, 38)( 11, 40)
( 12, 39)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)
( 20, 31)( 21, 25)( 22, 26)( 23, 28)( 24, 27)( 45,265)( 46,266)( 47,268)
( 48,267)( 49,305)( 50,306)( 51,308)( 52,307)( 53,301)( 54,302)( 55,304)
( 56,303)( 57,297)( 58,298)( 59,300)( 60,299)( 61,293)( 62,294)( 63,296)
( 64,295)( 65,289)( 66,290)( 67,292)( 68,291)( 69,285)( 70,286)( 71,288)
( 72,287)( 73,281)( 74,282)( 75,284)( 76,283)( 77,277)( 78,278)( 79,280)
( 80,279)( 81,273)( 82,274)( 83,276)( 84,275)( 85,269)( 86,270)( 87,272)
( 88,271)( 89,221)( 90,222)( 91,224)( 92,223)( 93,261)( 94,262)( 95,264)
( 96,263)( 97,257)( 98,258)( 99,260)(100,259)(101,253)(102,254)(103,256)
(104,255)(105,249)(106,250)(107,252)(108,251)(109,245)(110,246)(111,248)
(112,247)(113,241)(114,242)(115,244)(116,243)(117,237)(118,238)(119,240)
(120,239)(121,233)(122,234)(123,236)(124,235)(125,229)(126,230)(127,232)
(128,231)(129,225)(130,226)(131,228)(132,227)(133,177)(134,178)(135,180)
(136,179)(137,217)(138,218)(139,220)(140,219)(141,213)(142,214)(143,216)
(144,215)(145,209)(146,210)(147,212)(148,211)(149,205)(150,206)(151,208)
(152,207)(153,201)(154,202)(155,204)(156,203)(157,197)(158,198)(159,200)
(160,199)(161,193)(162,194)(163,196)(164,195)(165,189)(166,190)(167,192)
(168,191)(169,185)(170,186)(171,188)(172,187)(173,181)(174,182)(175,184)
(176,183);;
s2 := (  1, 49)(  2, 52)(  3, 51)(  4, 50)(  5, 45)(  6, 48)(  7, 47)(  8, 46)
(  9, 85)( 10, 88)( 11, 87)( 12, 86)( 13, 81)( 14, 84)( 15, 83)( 16, 82)
( 17, 77)( 18, 80)( 19, 79)( 20, 78)( 21, 73)( 22, 76)( 23, 75)( 24, 74)
( 25, 69)( 26, 72)( 27, 71)( 28, 70)( 29, 65)( 30, 68)( 31, 67)( 32, 66)
( 33, 61)( 34, 64)( 35, 63)( 36, 62)( 37, 57)( 38, 60)( 39, 59)( 40, 58)
( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 89,269)( 90,272)( 91,271)( 92,270)
( 93,265)( 94,268)( 95,267)( 96,266)( 97,305)( 98,308)( 99,307)(100,306)
(101,301)(102,304)(103,303)(104,302)(105,297)(106,300)(107,299)(108,298)
(109,293)(110,296)(111,295)(112,294)(113,289)(114,292)(115,291)(116,290)
(117,285)(118,288)(119,287)(120,286)(121,281)(122,284)(123,283)(124,282)
(125,277)(126,280)(127,279)(128,278)(129,273)(130,276)(131,275)(132,274)
(133,225)(134,228)(135,227)(136,226)(137,221)(138,224)(139,223)(140,222)
(141,261)(142,264)(143,263)(144,262)(145,257)(146,260)(147,259)(148,258)
(149,253)(150,256)(151,255)(152,254)(153,249)(154,252)(155,251)(156,250)
(157,245)(158,248)(159,247)(160,246)(161,241)(162,244)(163,243)(164,242)
(165,237)(166,240)(167,239)(168,238)(169,233)(170,236)(171,235)(172,234)
(173,229)(174,232)(175,231)(176,230)(177,181)(178,184)(179,183)(180,182)
(185,217)(186,220)(187,219)(188,218)(189,213)(190,216)(191,215)(192,214)
(193,209)(194,212)(195,211)(196,210)(197,205)(198,208)(199,207)(200,206)
(202,204);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope