Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1944a
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 81, 486, 81
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 9
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
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) :
   27-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope