Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,90}

Atlas Canonical Name {6,90}*1620b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1620,137)
Rank
3
Schläfli Type
{6,90}
Vertices, edges, …
9, 405, 135
Order of s0s1s2
45
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

3-fold

5-fold

15-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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 46, 91)( 47, 92)( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96)( 55,100)( 56,101)( 57,102)( 58,106)( 59,107)( 60,108)( 61,103)( 62,104)( 63,105)( 64,109)( 65,110)( 66,111)( 67,115)( 68,116)( 69,117)( 70,112)( 71,113)( 72,114)( 73,118)( 74,119)( 75,120)( 76,124)( 77,125)( 78,126)( 79,121)( 80,122)( 81,123)( 82,127)( 83,128)( 84,129)( 85,133)( 86,134)( 87,135)( 88,130)( 89,131)( 90,132)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(181,226)(182,227)(183,228)(184,232)(185,233)(186,234)(187,229)(188,230)(189,231)(190,235)(191,236)(192,237)(193,241)(194,242)(195,243)(196,238)(197,239)(198,240)(199,244)(200,245)(201,246)(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)(208,253)(209,254)(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)(217,262)(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)(224,266)(225,267)(274,277)(275,278)(276,279)(283,286)(284,287)(285,288)(292,295)(293,296)(294,297)(301,304)(302,305)(303,306)(310,313)(311,314)(312,315)(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)(323,365)(324,366)(325,370)(326,371)(327,372)(328,376)(329,377)(330,378)(331,373)(332,374)(333,375)(334,379)(335,380)(336,381)(337,385)(338,386)(339,387)(340,382)(341,383)(342,384)(343,388)(344,389)(345,390)(346,394)(347,395)(348,396)(349,391)(350,392)(351,393)(352,397)(353,398)(354,399)(355,403)(356,404)(357,405)(358,400)(359,401)(360,402);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 50)(  5, 49)(  6, 51)(  7, 54)(  8, 53)(  9, 52)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)( 17, 89)( 18, 88)( 19, 73)( 20, 75)( 21, 74)( 22, 77)( 23, 76)( 24, 78)( 25, 81)( 26, 80)( 27, 79)( 28, 64)( 29, 66)( 30, 65)( 31, 68)( 32, 67)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 37, 55)( 38, 57)( 39, 56)( 40, 59)( 41, 58)( 42, 60)( 43, 63)( 44, 62)( 45, 61)( 92, 93)( 94, 95)( 97, 99)(100,127)(101,129)(102,128)(103,131)(104,130)(105,132)(106,135)(107,134)(108,133)(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)(115,126)(116,125)(117,124)(136,317)(137,316)(138,318)(139,321)(140,320)(141,319)(142,322)(143,324)(144,323)(145,353)(146,352)(147,354)(148,357)(149,356)(150,355)(151,358)(152,360)(153,359)(154,344)(155,343)(156,345)(157,348)(158,347)(159,346)(160,349)(161,351)(162,350)(163,335)(164,334)(165,336)(166,339)(167,338)(168,337)(169,340)(170,342)(171,341)(172,326)(173,325)(174,327)(175,330)(176,329)(177,328)(178,331)(179,333)(180,332)(181,272)(182,271)(183,273)(184,276)(185,275)(186,274)(187,277)(188,279)(189,278)(190,308)(191,307)(192,309)(193,312)(194,311)(195,310)(196,313)(197,315)(198,314)(199,299)(200,298)(201,300)(202,303)(203,302)(204,301)(205,304)(206,306)(207,305)(208,290)(209,289)(210,291)(211,294)(212,293)(213,292)(214,295)(215,297)(216,296)(217,281)(218,280)(219,282)(220,285)(221,284)(222,283)(223,286)(224,288)(225,287)(226,362)(227,361)(228,363)(229,366)(230,365)(231,364)(232,367)(233,369)(234,368)(235,398)(236,397)(237,399)(238,402)(239,401)(240,400)(241,403)(242,405)(243,404)(244,389)(245,388)(246,390)(247,393)(248,392)(249,391)(250,394)(251,396)(252,395)(253,380)(254,379)(255,381)(256,384)(257,383)(258,382)(259,385)(260,387)(261,386)(262,371)(263,370)(264,372)(265,375)(266,374)(267,373)(268,376)(269,378)(270,377);;
s2 := (  1,145)(  2,147)(  3,146)(  4,151)(  5,153)(  6,152)(  7,148)(  8,150)(  9,149)( 10,136)( 11,138)( 12,137)( 13,142)( 14,144)( 15,143)( 16,139)( 17,141)( 18,140)( 19,172)( 20,174)( 21,173)( 22,178)( 23,180)( 24,179)( 25,175)( 26,177)( 27,176)( 28,163)( 29,165)( 30,164)( 31,169)( 32,171)( 33,170)( 34,166)( 35,168)( 36,167)( 37,154)( 38,156)( 39,155)( 40,160)( 41,162)( 42,161)( 43,157)( 44,159)( 45,158)( 46,195)( 47,194)( 48,193)( 49,192)( 50,191)( 51,190)( 52,198)( 53,197)( 54,196)( 55,186)( 56,185)( 57,184)( 58,183)( 59,182)( 60,181)( 61,189)( 62,188)( 63,187)( 64,222)( 65,221)( 66,220)( 67,219)( 68,218)( 69,217)( 70,225)( 71,224)( 72,223)( 73,213)( 74,212)( 75,211)( 76,210)( 77,209)( 78,208)( 79,216)( 80,215)( 81,214)( 82,204)( 83,203)( 84,202)( 85,201)( 86,200)( 87,199)( 88,207)( 89,206)( 90,205)( 91,243)( 92,242)( 93,241)( 94,240)( 95,239)( 96,238)( 97,237)( 98,236)( 99,235)(100,234)(101,233)(102,232)(103,231)(104,230)(105,229)(106,228)(107,227)(108,226)(109,270)(110,269)(111,268)(112,267)(113,266)(114,265)(115,264)(116,263)(117,262)(118,261)(119,260)(120,259)(121,258)(122,257)(123,256)(124,255)(125,254)(126,253)(127,252)(128,251)(129,250)(130,249)(131,248)(132,247)(133,246)(134,245)(135,244)(271,281)(272,280)(273,282)(274,287)(275,286)(276,288)(277,284)(278,283)(279,285)(289,308)(290,307)(291,309)(292,314)(293,313)(294,315)(295,311)(296,310)(297,312)(298,299)(301,305)(302,304)(303,306)(316,328)(317,330)(318,329)(319,325)(320,327)(321,326)(322,331)(323,333)(324,332)(334,355)(335,357)(336,356)(337,352)(338,354)(339,353)(340,358)(341,360)(342,359)(343,346)(344,348)(345,347)(350,351)(361,376)(362,378)(363,377)(364,373)(365,375)(366,374)(367,370)(368,372)(369,371)(379,403)(380,405)(381,404)(382,400)(383,402)(384,401)(385,397)(386,399)(387,398)(388,394)(389,396)(390,395)(392,393);;
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*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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(405)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 46, 91)( 47, 92)( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96)( 55,100)( 56,101)( 57,102)( 58,106)( 59,107)( 60,108)( 61,103)( 62,104)( 63,105)( 64,109)( 65,110)( 66,111)( 67,115)( 68,116)( 69,117)( 70,112)( 71,113)( 72,114)( 73,118)( 74,119)( 75,120)( 76,124)( 77,125)( 78,126)( 79,121)( 80,122)( 81,123)( 82,127)( 83,128)( 84,129)( 85,133)( 86,134)( 87,135)( 88,130)( 89,131)( 90,132)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(181,226)(182,227)(183,228)(184,232)(185,233)(186,234)(187,229)(188,230)(189,231)(190,235)(191,236)(192,237)(193,241)(194,242)(195,243)(196,238)(197,239)(198,240)(199,244)(200,245)(201,246)(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)(208,253)(209,254)(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)(217,262)(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)(224,266)(225,267)(274,277)(275,278)(276,279)(283,286)(284,287)(285,288)(292,295)(293,296)(294,297)(301,304)(302,305)(303,306)(310,313)(311,314)(312,315)(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)(323,365)(324,366)(325,370)(326,371)(327,372)(328,376)(329,377)(330,378)(331,373)(332,374)(333,375)(334,379)(335,380)(336,381)(337,385)(338,386)(339,387)(340,382)(341,383)(342,384)(343,388)(344,389)(345,390)(346,394)(347,395)(348,396)(349,391)(350,392)(351,393)(352,397)(353,398)(354,399)(355,403)(356,404)(357,405)(358,400)(359,401)(360,402);
s1 := Sym(405)!(  1, 46)(  2, 48)(  3, 47)(  4, 50)(  5, 49)(  6, 51)(  7, 54)(  8, 53)(  9, 52)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)( 17, 89)( 18, 88)( 19, 73)( 20, 75)( 21, 74)( 22, 77)( 23, 76)( 24, 78)( 25, 81)( 26, 80)( 27, 79)( 28, 64)( 29, 66)( 30, 65)( 31, 68)( 32, 67)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 37, 55)( 38, 57)( 39, 56)( 40, 59)( 41, 58)( 42, 60)( 43, 63)( 44, 62)( 45, 61)( 92, 93)( 94, 95)( 97, 99)(100,127)(101,129)(102,128)(103,131)(104,130)(105,132)(106,135)(107,134)(108,133)(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)(115,126)(116,125)(117,124)(136,317)(137,316)(138,318)(139,321)(140,320)(141,319)(142,322)(143,324)(144,323)(145,353)(146,352)(147,354)(148,357)(149,356)(150,355)(151,358)(152,360)(153,359)(154,344)(155,343)(156,345)(157,348)(158,347)(159,346)(160,349)(161,351)(162,350)(163,335)(164,334)(165,336)(166,339)(167,338)(168,337)(169,340)(170,342)(171,341)(172,326)(173,325)(174,327)(175,330)(176,329)(177,328)(178,331)(179,333)(180,332)(181,272)(182,271)(183,273)(184,276)(185,275)(186,274)(187,277)(188,279)(189,278)(190,308)(191,307)(192,309)(193,312)(194,311)(195,310)(196,313)(197,315)(198,314)(199,299)(200,298)(201,300)(202,303)(203,302)(204,301)(205,304)(206,306)(207,305)(208,290)(209,289)(210,291)(211,294)(212,293)(213,292)(214,295)(215,297)(216,296)(217,281)(218,280)(219,282)(220,285)(221,284)(222,283)(223,286)(224,288)(225,287)(226,362)(227,361)(228,363)(229,366)(230,365)(231,364)(232,367)(233,369)(234,368)(235,398)(236,397)(237,399)(238,402)(239,401)(240,400)(241,403)(242,405)(243,404)(244,389)(245,388)(246,390)(247,393)(248,392)(249,391)(250,394)(251,396)(252,395)(253,380)(254,379)(255,381)(256,384)(257,383)(258,382)(259,385)(260,387)(261,386)(262,371)(263,370)(264,372)(265,375)(266,374)(267,373)(268,376)(269,378)(270,377);
s2 := Sym(405)!(  1,145)(  2,147)(  3,146)(  4,151)(  5,153)(  6,152)(  7,148)(  8,150)(  9,149)( 10,136)( 11,138)( 12,137)( 13,142)( 14,144)( 15,143)( 16,139)( 17,141)( 18,140)( 19,172)( 20,174)( 21,173)( 22,178)( 23,180)( 24,179)( 25,175)( 26,177)( 27,176)( 28,163)( 29,165)( 30,164)( 31,169)( 32,171)( 33,170)( 34,166)( 35,168)( 36,167)( 37,154)( 38,156)( 39,155)( 40,160)( 41,162)( 42,161)( 43,157)( 44,159)( 45,158)( 46,195)( 47,194)( 48,193)( 49,192)( 50,191)( 51,190)( 52,198)( 53,197)( 54,196)( 55,186)( 56,185)( 57,184)( 58,183)( 59,182)( 60,181)( 61,189)( 62,188)( 63,187)( 64,222)( 65,221)( 66,220)( 67,219)( 68,218)( 69,217)( 70,225)( 71,224)( 72,223)( 73,213)( 74,212)( 75,211)( 76,210)( 77,209)( 78,208)( 79,216)( 80,215)( 81,214)( 82,204)( 83,203)( 84,202)( 85,201)( 86,200)( 87,199)( 88,207)( 89,206)( 90,205)( 91,243)( 92,242)( 93,241)( 94,240)( 95,239)( 96,238)( 97,237)( 98,236)( 99,235)(100,234)(101,233)(102,232)(103,231)(104,230)(105,229)(106,228)(107,227)(108,226)(109,270)(110,269)(111,268)(112,267)(113,266)(114,265)(115,264)(116,263)(117,262)(118,261)(119,260)(120,259)(121,258)(122,257)(123,256)(124,255)(125,254)(126,253)(127,252)(128,251)(129,250)(130,249)(131,248)(132,247)(133,246)(134,245)(135,244)(271,281)(272,280)(273,282)(274,287)(275,286)(276,288)(277,284)(278,283)(279,285)(289,308)(290,307)(291,309)(292,314)(293,313)(294,315)(295,311)(296,310)(297,312)(298,299)(301,305)(302,304)(303,306)(316,328)(317,330)(318,329)(319,325)(320,327)(321,326)(322,331)(323,333)(324,332)(334,355)(335,357)(336,356)(337,352)(338,354)(339,353)(340,358)(341,360)(342,359)(343,346)(344,348)(345,347)(350,351)(361,376)(362,378)(363,377)(364,373)(365,375)(366,374)(367,370)(368,372)(369,371)(379,403)(380,405)(381,404)(382,400)(383,402)(384,401)(385,397)(386,399)(387,398)(388,394)(389,396)(390,395)(392,393);
poly := sub<Sym(405)|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*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.

Twisty Puzzle