Polytope of Type {6,108}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,108}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2327)
Rank : 3
Schlafli Type : {6,108}
Number of vertices, edges, etc : 9, 486, 162
Order of s₀s₁s₂ : 108
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,36}*648a
   9-fold quotients : {6,12}*216c
   27-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 82)(  2, 84)(  3, 83)(  4, 89)(  5, 88)(  6, 90)(  7, 86)(  8, 85)
(  9, 87)( 10,104)( 11,103)( 12,105)( 13,101)( 14,100)( 15,102)( 16,108)
( 17,107)( 18,106)( 19, 95)( 20, 94)( 21, 96)( 22, 92)( 23, 91)( 24, 93)
( 25, 99)( 26, 98)( 27, 97)( 28,110)( 29,109)( 30,111)( 31,117)( 32,116)
( 33,115)( 34,114)( 35,113)( 36,112)( 37,132)( 38,131)( 39,130)( 40,129)
( 41,128)( 42,127)( 43,133)( 44,135)( 45,134)( 46,123)( 47,122)( 48,121)
( 49,120)( 50,119)( 51,118)( 52,124)( 53,126)( 54,125)( 55,138)( 56,137)
( 57,136)( 58,142)( 59,144)( 60,143)( 61,139)( 62,141)( 63,140)( 64,157)
( 65,159)( 66,158)( 67,154)( 68,156)( 69,155)( 70,161)( 71,160)( 72,162)
( 73,148)( 74,150)( 75,149)( 76,145)( 77,147)( 78,146)( 79,152)( 80,151)
( 81,153)(164,165)(166,170)(167,169)(168,171)(172,185)(173,184)(174,186)
(175,182)(176,181)(177,183)(178,189)(179,188)(180,187)(190,191)(193,198)
(194,197)(195,196)(199,213)(200,212)(201,211)(202,210)(203,209)(204,208)
(205,214)(206,216)(207,215)(217,219)(220,223)(221,225)(222,224)(226,238)
(227,240)(228,239)(229,235)(230,237)(231,236)(232,242)(233,241)(234,243);;
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, 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);;
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*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1 ];;
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, 82)(  2, 84)(  3, 83)(  4, 89)(  5, 88)(  6, 90)(  7, 86)
(  8, 85)(  9, 87)( 10,104)( 11,103)( 12,105)( 13,101)( 14,100)( 15,102)
( 16,108)( 17,107)( 18,106)( 19, 95)( 20, 94)( 21, 96)( 22, 92)( 23, 91)
( 24, 93)( 25, 99)( 26, 98)( 27, 97)( 28,110)( 29,109)( 30,111)( 31,117)
( 32,116)( 33,115)( 34,114)( 35,113)( 36,112)( 37,132)( 38,131)( 39,130)
( 40,129)( 41,128)( 42,127)( 43,133)( 44,135)( 45,134)( 46,123)( 47,122)
( 48,121)( 49,120)( 50,119)( 51,118)( 52,124)( 53,126)( 54,125)( 55,138)
( 56,137)( 57,136)( 58,142)( 59,144)( 60,143)( 61,139)( 62,141)( 63,140)
( 64,157)( 65,159)( 66,158)( 67,154)( 68,156)( 69,155)( 70,161)( 71,160)
( 72,162)( 73,148)( 74,150)( 75,149)( 76,145)( 77,147)( 78,146)( 79,152)
( 80,151)( 81,153)(164,165)(166,170)(167,169)(168,171)(172,185)(173,184)
(174,186)(175,182)(176,181)(177,183)(178,189)(179,188)(180,187)(190,191)
(193,198)(194,197)(195,196)(199,213)(200,212)(201,211)(202,210)(203,209)
(204,208)(205,214)(206,216)(207,215)(217,219)(220,223)(221,225)(222,224)
(226,238)(227,240)(228,239)(229,235)(230,237)(231,236)(232,242)(233,241)
(234,243);
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, 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);
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*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1 >;

References : None.
to this polytope