Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,219}

Atlas Canonical Name {4,219}*1752

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1752,51)
Rank
3
Schläfli Type
{4,219}
Vertices, edges, …
4, 438, 219
Order of s0s1s2
219
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

73-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

Twisty Puzzle