Polytope of Type {18,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*1944ae
if this polytope has a name.
Group : SmallGroup(1944,2339)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
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) :
   3-fold quotients : {18,18}*648b, {6,18}*648i
   6-fold quotients : {18,9}*324
   9-fold quotients : {6,18}*216a, {6,18}*216b, {6,6}*216d
   18-fold quotients : {6,9}*108
   27-fold quotients : {2,18}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {2,9}*36, {3,6}*36, {6,3}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope