Part of the Atlas of Small Regular Polytopes

Polytope of Type {81,6}

Atlas Canonical Name {81,6}*972

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(972,114)
Rank
3
Schläfli Type
{81,6}
Vertices, edges, …
81, 243, 6
Order of s0s1s2
162
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

81-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

Twisty Puzzle