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Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1944f
if this polytope has a name.
Group : SmallGroup(1944,806)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 81, 486, 54
Order of s0s1s2 : 4
Order of s0s1s2s1 : 18
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) :
   9-fold quotients : {6,12}*216b
   27-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 17)( 11, 16)( 12, 18)( 13, 14)
( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 75)( 29, 74)( 30, 73)( 31, 81)
( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 76)( 37, 61)( 38, 63)( 39, 62)
( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 68)( 47, 67)
( 48, 69)( 49, 65)( 50, 64)( 51, 66)( 52, 71)( 53, 70)( 54, 72)( 82,168)
( 83,167)( 84,166)( 85,165)( 86,164)( 87,163)( 88,171)( 89,170)( 90,169)
( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)( 98,177)
( 99,176)(100,188)(101,187)(102,189)(103,185)(104,184)(105,186)(106,182)
(107,181)(108,183)(109,239)(110,238)(111,240)(112,236)(113,235)(114,237)
(115,242)(116,241)(117,243)(118,219)(119,218)(120,217)(121,225)(122,224)
(123,223)(124,222)(125,221)(126,220)(127,232)(128,234)(129,233)(130,229)
(131,231)(132,230)(133,226)(134,228)(135,227)(136,201)(137,200)(138,199)
(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,214)(146,216)
(147,215)(148,211)(149,213)(150,212)(151,208)(152,210)(153,209)(154,194)
(155,193)(156,195)(157,191)(158,190)(159,192)(160,197)(161,196)(162,198);;
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)( 28,135)( 29,131)( 30,127)( 31,129)( 32,134)
( 33,130)( 34,132)( 35,128)( 36,133)( 37,123)( 38,119)( 39,124)( 40,126)
( 41,122)( 42,118)( 43,120)( 44,125)( 45,121)( 46,111)( 47,116)( 48,112)
( 49,114)( 50,110)( 51,115)( 52,117)( 53,113)( 54,109)( 55,147)( 56,152)
( 57,148)( 58,150)( 59,146)( 60,151)( 61,153)( 62,149)( 63,145)( 64,144)
( 65,140)( 66,136)( 67,138)( 68,143)( 69,139)( 70,141)( 71,137)( 72,142)
( 73,159)( 74,155)( 75,160)( 76,162)( 77,158)( 78,154)( 79,156)( 80,161)
( 81,157)(163,168)(165,169)(166,171)(172,183)(173,188)(174,184)(175,186)
(176,182)(177,187)(178,189)(179,185)(180,181)(190,209)(191,214)(192,213)
(193,212)(194,208)(195,216)(196,215)(197,211)(198,210)(199,206)(200,202)
(203,205)(217,230)(218,226)(219,234)(220,233)(221,229)(222,228)(223,227)
(224,232)(225,231)(235,242)(236,238)(239,241);;
s2 := (  1, 15)(  2, 14)(  3, 13)(  4, 17)(  5, 16)(  6, 18)(  7, 10)(  8, 12)
(  9, 11)( 19, 20)( 23, 24)( 25, 27)( 28,168)( 29,167)( 30,166)( 31,170)
( 32,169)( 33,171)( 34,163)( 35,165)( 36,164)( 37,187)( 38,189)( 39,188)
( 40,183)( 41,182)( 42,181)( 43,185)( 44,184)( 45,186)( 46,173)( 47,172)
( 48,174)( 49,175)( 50,177)( 51,176)( 52,180)( 53,179)( 54,178)( 55,103)
( 56,105)( 57,104)( 58,108)( 59,107)( 60,106)( 61,101)( 62,100)( 63,102)
( 64, 98)( 65, 97)( 66, 99)( 67, 91)( 68, 93)( 69, 92)( 70, 96)( 71, 95)
( 72, 94)( 73, 84)( 74, 83)( 75, 82)( 76, 86)( 77, 85)( 78, 87)( 79, 88)
( 80, 90)( 81, 89)(109,235)(110,237)(111,236)(112,240)(113,239)(114,238)
(115,242)(116,241)(117,243)(118,230)(119,229)(120,231)(121,232)(122,234)
(123,233)(124,228)(125,227)(126,226)(127,225)(128,224)(129,223)(130,218)
(131,217)(132,219)(133,220)(134,222)(135,221)(136,162)(137,161)(138,160)
(139,155)(140,154)(141,156)(142,157)(143,159)(144,158)(146,147)(148,150)
(151,152)(190,202)(191,204)(192,203)(193,207)(194,206)(195,205)(196,200)
(197,199)(198,201)(208,210)(211,212)(215,216);;
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*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(243)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 17)( 11, 16)( 12, 18)
( 13, 14)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 75)( 29, 74)( 30, 73)
( 31, 81)( 32, 80)( 33, 79)( 34, 78)( 35, 77)( 36, 76)( 37, 61)( 38, 63)
( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 68)
( 47, 67)( 48, 69)( 49, 65)( 50, 64)( 51, 66)( 52, 71)( 53, 70)( 54, 72)
( 82,168)( 83,167)( 84,166)( 85,165)( 86,164)( 87,163)( 88,171)( 89,170)
( 90,169)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)
( 98,177)( 99,176)(100,188)(101,187)(102,189)(103,185)(104,184)(105,186)
(106,182)(107,181)(108,183)(109,239)(110,238)(111,240)(112,236)(113,235)
(114,237)(115,242)(116,241)(117,243)(118,219)(119,218)(120,217)(121,225)
(122,224)(123,223)(124,222)(125,221)(126,220)(127,232)(128,234)(129,233)
(130,229)(131,231)(132,230)(133,226)(134,228)(135,227)(136,201)(137,200)
(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,214)
(146,216)(147,215)(148,211)(149,213)(150,212)(151,208)(152,210)(153,209)
(154,194)(155,193)(156,195)(157,191)(158,190)(159,192)(160,197)(161,196)
(162,198);
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)( 28,135)( 29,131)( 30,127)( 31,129)
( 32,134)( 33,130)( 34,132)( 35,128)( 36,133)( 37,123)( 38,119)( 39,124)
( 40,126)( 41,122)( 42,118)( 43,120)( 44,125)( 45,121)( 46,111)( 47,116)
( 48,112)( 49,114)( 50,110)( 51,115)( 52,117)( 53,113)( 54,109)( 55,147)
( 56,152)( 57,148)( 58,150)( 59,146)( 60,151)( 61,153)( 62,149)( 63,145)
( 64,144)( 65,140)( 66,136)( 67,138)( 68,143)( 69,139)( 70,141)( 71,137)
( 72,142)( 73,159)( 74,155)( 75,160)( 76,162)( 77,158)( 78,154)( 79,156)
( 80,161)( 81,157)(163,168)(165,169)(166,171)(172,183)(173,188)(174,184)
(175,186)(176,182)(177,187)(178,189)(179,185)(180,181)(190,209)(191,214)
(192,213)(193,212)(194,208)(195,216)(196,215)(197,211)(198,210)(199,206)
(200,202)(203,205)(217,230)(218,226)(219,234)(220,233)(221,229)(222,228)
(223,227)(224,232)(225,231)(235,242)(236,238)(239,241);
s2 := Sym(243)!(  1, 15)(  2, 14)(  3, 13)(  4, 17)(  5, 16)(  6, 18)(  7, 10)
(  8, 12)(  9, 11)( 19, 20)( 23, 24)( 25, 27)( 28,168)( 29,167)( 30,166)
( 31,170)( 32,169)( 33,171)( 34,163)( 35,165)( 36,164)( 37,187)( 38,189)
( 39,188)( 40,183)( 41,182)( 42,181)( 43,185)( 44,184)( 45,186)( 46,173)
( 47,172)( 48,174)( 49,175)( 50,177)( 51,176)( 52,180)( 53,179)( 54,178)
( 55,103)( 56,105)( 57,104)( 58,108)( 59,107)( 60,106)( 61,101)( 62,100)
( 63,102)( 64, 98)( 65, 97)( 66, 99)( 67, 91)( 68, 93)( 69, 92)( 70, 96)
( 71, 95)( 72, 94)( 73, 84)( 74, 83)( 75, 82)( 76, 86)( 77, 85)( 78, 87)
( 79, 88)( 80, 90)( 81, 89)(109,235)(110,237)(111,236)(112,240)(113,239)
(114,238)(115,242)(116,241)(117,243)(118,230)(119,229)(120,231)(121,232)
(122,234)(123,233)(124,228)(125,227)(126,226)(127,225)(128,224)(129,223)
(130,218)(131,217)(132,219)(133,220)(134,222)(135,221)(136,162)(137,161)
(138,160)(139,155)(140,154)(141,156)(142,157)(143,159)(144,158)(146,147)
(148,150)(151,152)(190,202)(191,204)(192,203)(193,207)(194,206)(195,205)
(196,200)(197,199)(198,201)(208,210)(211,212)(215,216);
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*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0 >; 
 
References : None.
to this polytope