Polytope of Type {18,12}

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

References : None.
to this polytope