Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1944f
if this polytope has a name.
Group : SmallGroup(1944,2325)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 81, 486, 81
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 9
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   27-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope