Questions?
See the FAQ
or other info.

Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2322)
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
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,4}*648
   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, 85)(  2, 87)(  3, 86)(  4, 89)(  5, 88)(  6, 90)(  7, 84)(  8, 83)
(  9, 82)( 10,103)( 11,105)( 12,104)( 13,107)( 14,106)( 15,108)( 16,102)
( 17,101)( 18,100)( 19, 94)( 20, 96)( 21, 95)( 22, 98)( 23, 97)( 24, 99)
( 25, 93)( 26, 92)( 27, 91)( 28,114)( 29,113)( 30,112)( 31,115)( 32,117)
( 33,116)( 34,110)( 35,109)( 36,111)( 37,132)( 38,131)( 39,130)( 40,133)
( 41,135)( 42,134)( 43,128)( 44,127)( 45,129)( 46,123)( 47,122)( 48,121)
( 49,124)( 50,126)( 51,125)( 52,119)( 53,118)( 54,120)( 55,140)( 56,139)
( 57,141)( 58,144)( 59,143)( 60,142)( 61,136)( 62,138)( 63,137)( 64,158)
( 65,157)( 66,159)( 67,162)( 68,161)( 69,160)( 70,154)( 71,156)( 72,155)
( 73,149)( 74,148)( 75,150)( 76,153)( 77,152)( 78,151)( 79,145)( 80,147)
( 81,146)(163,165)(167,168)(169,170)(172,183)(173,182)(174,181)(175,184)
(176,186)(177,185)(178,188)(179,187)(180,189)(190,191)(193,195)(197,198)
(199,209)(200,208)(201,210)(202,213)(203,212)(204,211)(205,214)(206,216)
(207,215)(218,219)(220,221)(223,225)(226,235)(227,237)(228,236)(229,239)
(230,238)(231,240)(232,243)(233,242)(234,241);;
s2 := (  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);;
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*s0*s1*s2*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2 ];;
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, 85)(  2, 87)(  3, 86)(  4, 89)(  5, 88)(  6, 90)(  7, 84)
(  8, 83)(  9, 82)( 10,103)( 11,105)( 12,104)( 13,107)( 14,106)( 15,108)
( 16,102)( 17,101)( 18,100)( 19, 94)( 20, 96)( 21, 95)( 22, 98)( 23, 97)
( 24, 99)( 25, 93)( 26, 92)( 27, 91)( 28,114)( 29,113)( 30,112)( 31,115)
( 32,117)( 33,116)( 34,110)( 35,109)( 36,111)( 37,132)( 38,131)( 39,130)
( 40,133)( 41,135)( 42,134)( 43,128)( 44,127)( 45,129)( 46,123)( 47,122)
( 48,121)( 49,124)( 50,126)( 51,125)( 52,119)( 53,118)( 54,120)( 55,140)
( 56,139)( 57,141)( 58,144)( 59,143)( 60,142)( 61,136)( 62,138)( 63,137)
( 64,158)( 65,157)( 66,159)( 67,162)( 68,161)( 69,160)( 70,154)( 71,156)
( 72,155)( 73,149)( 74,148)( 75,150)( 76,153)( 77,152)( 78,151)( 79,145)
( 80,147)( 81,146)(163,165)(167,168)(169,170)(172,183)(173,182)(174,181)
(175,184)(176,186)(177,185)(178,188)(179,187)(180,189)(190,191)(193,195)
(197,198)(199,209)(200,208)(201,210)(202,213)(203,212)(204,211)(205,214)
(206,216)(207,215)(218,219)(220,221)(223,225)(226,235)(227,237)(228,236)
(229,239)(230,238)(231,240)(232,243)(233,242)(234,241);
s2 := 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);
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*s0*s1*s2*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2 >; 
 
References : None.
to this polytope