Polytope of Type {108,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {108,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,2326)
Rank : 3
Schlafli Type : {108,6}
Number of vertices, edges, etc : 162, 486, 9
Order of s0s1s2 : 108
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {36,6}*648a
   9-fold quotients : {12,6}*216c
   27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)( 32, 34)( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)( 82,163)( 83,165)( 84,164)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,189)( 98,188)( 99,187)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,180)(107,179)(108,178)(109,190)(110,192)(111,191)(112,197)(113,196)(114,198)(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)(122,208)(123,210)(124,216)(125,215)(126,214)(127,203)(128,202)(129,204)(130,200)(131,199)(132,201)(133,207)(134,206)(135,205)(136,217)(137,219)(138,218)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,243)(152,242)(153,241)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,234)(161,233)(162,232);;
s1 := (  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, 91)( 29, 93)( 30, 92)( 31, 98)( 32, 97)( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 82)( 38, 84)( 39, 83)( 40, 89)( 41, 88)( 42, 90)( 43, 86)( 44, 85)( 45, 87)( 46,104)( 47,103)( 48,105)( 49,101)( 50,100)( 51,102)( 52,108)( 53,107)( 54,106)( 55,172)( 56,174)( 57,173)( 58,179)( 59,178)( 60,180)( 61,176)( 62,175)( 63,177)( 64,163)( 65,165)( 66,164)( 67,170)( 68,169)( 69,171)( 70,167)( 71,166)( 72,168)( 73,185)( 74,184)( 75,186)( 76,182)( 77,181)( 78,183)( 79,189)( 80,188)( 81,187)(109,118)(110,120)(111,119)(112,125)(113,124)(114,126)(115,122)(116,121)(117,123)(127,131)(128,130)(129,132)(133,135)(136,199)(137,201)(138,200)(139,206)(140,205)(141,207)(142,203)(143,202)(144,204)(145,190)(146,192)(147,191)(148,197)(149,196)(150,198)(151,194)(152,193)(153,195)(154,212)(155,211)(156,213)(157,209)(158,208)(159,210)(160,216)(161,215)(162,214)(217,226)(218,228)(219,227)(220,233)(221,232)(222,234)(223,230)(224,229)(225,231)(235,239)(236,238)(237,240)(241,243);;
s2 := (  1, 28)(  2, 29)(  3, 30)(  4, 31)(  5, 32)(  6, 33)(  7, 34)(  8, 35)(  9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)( 16, 43)( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)( 24, 51)( 25, 52)( 26, 53)( 27, 54)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)(162,243);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(243)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 10, 23)( 11, 22)( 12, 24)( 13, 20)( 14, 19)( 15, 21)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 35)( 32, 34)( 33, 36)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)( 82,163)( 83,165)( 84,164)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,189)( 98,188)( 99,187)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,180)(107,179)(108,178)(109,190)(110,192)(111,191)(112,197)(113,196)(114,198)(115,194)(116,193)(117,195)(118,212)(119,211)(120,213)(121,209)(122,208)(123,210)(124,216)(125,215)(126,214)(127,203)(128,202)(129,204)(130,200)(131,199)(132,201)(133,207)(134,206)(135,205)(136,217)(137,219)(138,218)(139,224)(140,223)(141,225)(142,221)(143,220)(144,222)(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,243)(152,242)(153,241)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,234)(161,233)(162,232);
s1 := 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, 91)( 29, 93)( 30, 92)( 31, 98)( 32, 97)( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 82)( 38, 84)( 39, 83)( 40, 89)( 41, 88)( 42, 90)( 43, 86)( 44, 85)( 45, 87)( 46,104)( 47,103)( 48,105)( 49,101)( 50,100)( 51,102)( 52,108)( 53,107)( 54,106)( 55,172)( 56,174)( 57,173)( 58,179)( 59,178)( 60,180)( 61,176)( 62,175)( 63,177)( 64,163)( 65,165)( 66,164)( 67,170)( 68,169)( 69,171)( 70,167)( 71,166)( 72,168)( 73,185)( 74,184)( 75,186)( 76,182)( 77,181)( 78,183)( 79,189)( 80,188)( 81,187)(109,118)(110,120)(111,119)(112,125)(113,124)(114,126)(115,122)(116,121)(117,123)(127,131)(128,130)(129,132)(133,135)(136,199)(137,201)(138,200)(139,206)(140,205)(141,207)(142,203)(143,202)(144,204)(145,190)(146,192)(147,191)(148,197)(149,196)(150,198)(151,194)(152,193)(153,195)(154,212)(155,211)(156,213)(157,209)(158,208)(159,210)(160,216)(161,215)(162,214)(217,226)(218,228)(219,227)(220,233)(221,232)(222,234)(223,230)(224,229)(225,231)(235,239)(236,238)(237,240)(241,243);
s2 := Sym(243)!(  1, 28)(  2, 29)(  3, 30)(  4, 31)(  5, 32)(  6, 33)(  7, 34)(  8, 35)(  9, 36)( 10, 37)( 11, 38)( 12, 39)( 13, 40)( 14, 41)( 15, 42)( 16, 43)( 17, 44)( 18, 45)( 19, 46)( 20, 47)( 21, 48)( 22, 49)( 23, 50)( 24, 51)( 25, 52)( 26, 53)( 27, 54)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)(105,213)(106,214)(107,215)(108,216)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,217)(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)(162,243);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s2*s0*s1*s2*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*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*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 >;
References : None.
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