Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1944
if this polytope has a name.
Group : SmallGroup(1944,2323)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 162, 486, 27
Order of s₀s₁s₂ : 36
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, {12,6}*648, {36,6}*648b, {36,6}*648c
   9-fold quotients : {4,6}*216, {12,6}*216a, {12,6}*216b, {12,6}*216c
   27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope