Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*972d
if this polytope has a name.
Group : SmallGroup(972,104)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 27, 243, 27
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,18,2} of size 1944
Vertex Figure Of :
   {2,18,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,18}*324b, {18,6}*324c
   9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,18}*1944i
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope