Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1944i
if this polytope has a name.
Group : SmallGroup(1944,2325)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 81, 486, 54
Order of s₀s₁s₂ : 12
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}*216c
   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)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 62)( 29, 61)( 30, 63)( 31, 59)
( 32, 58)( 33, 60)( 34, 56)( 35, 55)( 36, 57)( 37, 71)( 38, 70)( 39, 72)
( 40, 68)( 41, 67)( 42, 69)( 43, 65)( 44, 64)( 45, 66)( 46, 80)( 47, 79)
( 48, 81)( 49, 77)( 50, 76)( 51, 78)( 52, 74)( 53, 73)( 54, 75)( 82,164)
( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)
( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)( 96,180)( 97,176)( 98,175)
( 99,177)(100,182)(101,181)(102,183)(103,188)(104,187)(105,189)(106,185)
(107,184)(108,186)(109,225)(110,224)(111,223)(112,222)(113,221)(114,220)
(115,219)(116,218)(117,217)(118,234)(119,233)(120,232)(121,231)(122,230)
(123,229)(124,228)(125,227)(126,226)(127,243)(128,242)(129,241)(130,240)
(131,239)(132,238)(133,237)(134,236)(135,235)(136,198)(137,197)(138,196)
(139,195)(140,194)(141,193)(142,192)(143,191)(144,190)(145,207)(146,206)
(147,205)(148,204)(149,203)(150,202)(151,201)(152,200)(153,199)(154,216)
(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)(161,209)(162,208);;
s1 := (  1, 91)(  2, 93)(  3, 92)(  4, 95)(  5, 94)(  6, 96)(  7, 99)(  8, 98)
(  9, 97)( 10, 89)( 11, 88)( 12, 90)( 13, 84)( 14, 83)( 15, 82)( 16, 85)
( 17, 87)( 18, 86)( 19,105)( 20,104)( 21,103)( 22,106)( 23,108)( 24,107)
( 25,101)( 26,100)( 27,102)( 28,120)( 29,119)( 30,118)( 31,121)( 32,123)
( 33,122)( 34,125)( 35,124)( 36,126)( 37,115)( 38,117)( 39,116)( 40,110)
( 41,109)( 42,111)( 43,114)( 44,113)( 45,112)( 46,131)( 47,130)( 48,132)
( 49,135)( 50,134)( 51,133)( 52,127)( 53,129)( 54,128)( 55,146)( 56,145)
( 57,147)( 58,150)( 59,149)( 60,148)( 61,151)( 62,153)( 63,152)( 64,144)
( 65,143)( 66,142)( 67,136)( 68,138)( 69,137)( 70,140)( 71,139)( 72,141)
( 73,157)( 74,159)( 75,158)( 76,161)( 77,160)( 78,162)( 79,156)( 80,155)
( 81,154)(163,180)(164,179)(165,178)(166,172)(167,174)(168,173)(169,176)
(170,175)(171,177)(181,182)(184,186)(188,189)(190,206)(191,205)(192,207)
(193,201)(194,200)(195,199)(196,202)(197,204)(198,203)(209,210)(211,212)
(214,216)(217,232)(218,234)(219,233)(220,227)(221,226)(222,228)(223,231)
(224,230)(225,229)(235,237)(239,240)(241,242);;
s2 := (  2,  8)(  3,  6)(  4,  7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)( 14, 23)
( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 28, 88)( 29, 86)( 30, 84)( 31, 85)
( 32, 83)( 33, 90)( 34, 82)( 35, 89)( 36, 87)( 37,106)( 38,104)( 39,102)
( 40,103)( 41,101)( 42,108)( 43,100)( 44,107)( 45,105)( 46, 97)( 47, 95)
( 48, 93)( 49, 94)( 50, 92)( 51, 99)( 52, 91)( 53, 98)( 54, 96)( 55,166)
( 56,164)( 57,171)( 58,163)( 59,170)( 60,168)( 61,169)( 62,167)( 63,165)
( 64,184)( 65,182)( 66,189)( 67,181)( 68,188)( 69,186)( 70,187)( 71,185)
( 72,183)( 73,175)( 74,173)( 75,180)( 76,172)( 77,179)( 78,177)( 79,178)
( 80,176)( 81,174)(109,112)(111,117)(113,116)(118,130)(119,128)(120,135)
(121,127)(122,134)(123,132)(124,133)(125,131)(126,129)(136,190)(137,197)
(138,195)(139,196)(140,194)(141,192)(142,193)(143,191)(144,198)(145,208)
(146,215)(147,213)(148,214)(149,212)(150,210)(151,211)(152,209)(153,216)
(154,199)(155,206)(156,204)(157,205)(158,203)(159,201)(160,202)(161,200)
(162,207)(217,223)(218,221)(222,225)(226,241)(227,239)(228,237)(229,238)
(230,236)(231,243)(232,235)(233,242)(234,240);;
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope