Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,54}

Atlas Canonical Name {6,54}*1944g

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Overview

Group
SmallGroup(1944,2343)
Rank
3
Schläfli Type
{6,54}
Vertices, edges, …
18, 486, 162
Order of s0s1s2
54
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

6-fold

9-fold

18-fold

27-fold

54-fold

81-fold

162-fold

243-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*s0*(s2*s1)^3*s0*(s2*s1)^23*s2> of order 2

81 facets

12 vertex figures

P/N, where N=<(s0*s1)^2> of order 3

108 facets

6 vertex figures

P/N, where N=<s0*(s1*s2)^2*s1*s0*(s2*s1)^15*s2> of order 3

54 facets

6 vertex figures

P/N, where N=<s0*(s1*s2)^3*s1*s0*(s2*s1)^14*s2> of order 3

54 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle