Polytope of Type {4,213}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,213}*1704
if this polytope has a name.
Group : SmallGroup(1704,32)
Rank : 3
Schlafli Type : {4,213}
Number of vertices, edges, etc : 4, 426, 213
Order of s₀s₁s₂ : 213
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) :
   71-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);;
s1 := (  3,  4)(  5,281)(  6,282)(  7,284)(  8,283)(  9,277)( 10,278)( 11,280)
( 12,279)( 13,273)( 14,274)( 15,276)( 16,275)( 17,269)( 18,270)( 19,272)
( 20,271)( 21,265)( 22,266)( 23,268)( 24,267)( 25,261)( 26,262)( 27,264)
( 28,263)( 29,257)( 30,258)( 31,260)( 32,259)( 33,253)( 34,254)( 35,256)
( 36,255)( 37,249)( 38,250)( 39,252)( 40,251)( 41,245)( 42,246)( 43,248)
( 44,247)( 45,241)( 46,242)( 47,244)( 48,243)( 49,237)( 50,238)( 51,240)
( 52,239)( 53,233)( 54,234)( 55,236)( 56,235)( 57,229)( 58,230)( 59,232)
( 60,231)( 61,225)( 62,226)( 63,228)( 64,227)( 65,221)( 66,222)( 67,224)
( 68,223)( 69,217)( 70,218)( 71,220)( 72,219)( 73,213)( 74,214)( 75,216)
( 76,215)( 77,209)( 78,210)( 79,212)( 80,211)( 81,205)( 82,206)( 83,208)
( 84,207)( 85,201)( 86,202)( 87,204)( 88,203)( 89,197)( 90,198)( 91,200)
( 92,199)( 93,193)( 94,194)( 95,196)( 96,195)( 97,189)( 98,190)( 99,192)
(100,191)(101,185)(102,186)(103,188)(104,187)(105,181)(106,182)(107,184)
(108,183)(109,177)(110,178)(111,180)(112,179)(113,173)(114,174)(115,176)
(116,175)(117,169)(118,170)(119,172)(120,171)(121,165)(122,166)(123,168)
(124,167)(125,161)(126,162)(127,164)(128,163)(129,157)(130,158)(131,160)
(132,159)(133,153)(134,154)(135,156)(136,155)(137,149)(138,150)(139,152)
(140,151)(141,145)(142,146)(143,148)(144,147);;
s2 := (  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9,281)( 10,284)( 11,283)( 12,282)
( 13,277)( 14,280)( 15,279)( 16,278)( 17,273)( 18,276)( 19,275)( 20,274)
( 21,269)( 22,272)( 23,271)( 24,270)( 25,265)( 26,268)( 27,267)( 28,266)
( 29,261)( 30,264)( 31,263)( 32,262)( 33,257)( 34,260)( 35,259)( 36,258)
( 37,253)( 38,256)( 39,255)( 40,254)( 41,249)( 42,252)( 43,251)( 44,250)
( 45,245)( 46,248)( 47,247)( 48,246)( 49,241)( 50,244)( 51,243)( 52,242)
( 53,237)( 54,240)( 55,239)( 56,238)( 57,233)( 58,236)( 59,235)( 60,234)
( 61,229)( 62,232)( 63,231)( 64,230)( 65,225)( 66,228)( 67,227)( 68,226)
( 69,221)( 70,224)( 71,223)( 72,222)( 73,217)( 74,220)( 75,219)( 76,218)
( 77,213)( 78,216)( 79,215)( 80,214)( 81,209)( 82,212)( 83,211)( 84,210)
( 85,205)( 86,208)( 87,207)( 88,206)( 89,201)( 90,204)( 91,203)( 92,202)
( 93,197)( 94,200)( 95,199)( 96,198)( 97,193)( 98,196)( 99,195)(100,194)
(101,189)(102,192)(103,191)(104,190)(105,185)(106,188)(107,187)(108,186)
(109,181)(110,184)(111,183)(112,182)(113,177)(114,180)(115,179)(116,178)
(117,173)(118,176)(119,175)(120,174)(121,169)(122,172)(123,171)(124,170)
(125,165)(126,168)(127,167)(128,166)(129,161)(130,164)(131,163)(132,162)
(133,157)(134,160)(135,159)(136,158)(137,153)(138,156)(139,155)(140,154)
(141,149)(142,152)(143,151)(144,150)(146,148);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope