Polytope of Type {108,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {108,6}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2327)
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
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.
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)(109,119)(110,118)(111,120)(112,126)(113,125)
(114,124)(115,123)(116,122)(117,121)(127,132)(128,131)(129,130)(134,135)
(136,201)(137,200)(138,199)(139,205)(140,207)(141,206)(142,202)(143,204)
(144,203)(145,192)(146,191)(147,190)(148,196)(149,198)(150,197)(151,193)
(152,195)(153,194)(154,211)(155,213)(156,212)(157,208)(158,210)(159,209)
(160,215)(161,214)(162,216)(217,227)(218,226)(219,228)(220,234)(221,233)
(222,232)(223,231)(224,230)(225,229)(235,240)(236,239)(237,238)(242,243);;
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*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, 
s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0 ];;
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)(109,119)(110,118)(111,120)(112,126)
(113,125)(114,124)(115,123)(116,122)(117,121)(127,132)(128,131)(129,130)
(134,135)(136,201)(137,200)(138,199)(139,205)(140,207)(141,206)(142,202)
(143,204)(144,203)(145,192)(146,191)(147,190)(148,196)(149,198)(150,197)
(151,193)(152,195)(153,194)(154,211)(155,213)(156,212)(157,208)(158,210)
(159,209)(160,215)(161,214)(162,216)(217,227)(218,226)(219,228)(220,234)
(221,233)(222,232)(223,231)(224,230)(225,229)(235,240)(236,239)(237,238)
(242,243);
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*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, 
s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0 >;

References : None.
to this polytope