Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*972e
if this polytope has a name.
Group : SmallGroup(972,105)
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
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
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}*324b
9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944m
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, 57)( 29, 56)( 30, 55)( 31, 60)
( 32, 59)( 33, 58)( 34, 63)( 35, 62)( 36, 61)( 37, 78)( 38, 77)( 39, 76)
( 40, 81)( 41, 80)( 42, 79)( 43, 75)( 44, 74)( 45, 73)( 46, 72)( 47, 71)
( 48, 70)( 49, 66)( 50, 65)( 51, 64)( 52, 69)( 53, 68)( 54, 67)( 83, 84)
( 86, 87)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,106)( 95,108)( 96,107)
( 97,100)( 98,102)( 99,101)(109,138)(110,137)(111,136)(112,141)(113,140)
(114,139)(115,144)(116,143)(117,142)(118,159)(119,158)(120,157)(121,162)
(122,161)(123,160)(124,156)(125,155)(126,154)(127,153)(128,152)(129,151)
(130,147)(131,146)(132,145)(133,150)(134,149)(135,148)(164,165)(167,168)
(170,171)(172,184)(173,186)(174,185)(175,187)(176,189)(177,188)(178,181)
(179,183)(180,182)(190,219)(191,218)(192,217)(193,222)(194,221)(195,220)
(196,225)(197,224)(198,223)(199,240)(200,239)(201,238)(202,243)(203,242)
(204,241)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232)(211,228)
(212,227)(213,226)(214,231)(215,230)(216,229);;
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, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 65)
( 67, 71)( 68, 70)( 69, 72)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 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)(136,222)(137,221)(138,220)
(139,219)(140,218)(141,217)(142,225)(143,224)(144,223)(145,230)(146,229)
(147,231)(148,227)(149,226)(150,228)(151,233)(152,232)(153,234)(154,238)
(155,240)(156,239)(157,235)(158,237)(159,236)(160,241)(161,243)(162,242);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0 ];;
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, 57)( 29, 56)( 30, 55)
( 31, 60)( 32, 59)( 33, 58)( 34, 63)( 35, 62)( 36, 61)( 37, 78)( 38, 77)
( 39, 76)( 40, 81)( 41, 80)( 42, 79)( 43, 75)( 44, 74)( 45, 73)( 46, 72)
( 47, 71)( 48, 70)( 49, 66)( 50, 65)( 51, 64)( 52, 69)( 53, 68)( 54, 67)
( 83, 84)( 86, 87)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,106)( 95,108)
( 96,107)( 97,100)( 98,102)( 99,101)(109,138)(110,137)(111,136)(112,141)
(113,140)(114,139)(115,144)(116,143)(117,142)(118,159)(119,158)(120,157)
(121,162)(122,161)(123,160)(124,156)(125,155)(126,154)(127,153)(128,152)
(129,151)(130,147)(131,146)(132,145)(133,150)(134,149)(135,148)(164,165)
(167,168)(170,171)(172,184)(173,186)(174,185)(175,187)(176,189)(177,188)
(178,181)(179,183)(180,182)(190,219)(191,218)(192,217)(193,222)(194,221)
(195,220)(196,225)(197,224)(198,223)(199,240)(200,239)(201,238)(202,243)
(203,242)(204,241)(205,237)(206,236)(207,235)(208,234)(209,233)(210,232)
(211,228)(212,227)(213,226)(214,231)(215,230)(216,229);
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, 57)( 58, 63)( 59, 62)( 60, 61)
( 64, 65)( 67, 71)( 68, 70)( 69, 72)( 74, 75)( 76, 79)( 77, 81)( 78, 80)
( 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)(136,222)(137,221)
(138,220)(139,219)(140,218)(141,217)(142,225)(143,224)(144,223)(145,230)
(146,229)(147,231)(148,227)(149,226)(150,228)(151,233)(152,232)(153,234)
(154,238)(155,240)(156,239)(157,235)(158,237)(159,236)(160,241)(161,243)
(162,242);
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*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0 >;
References : None.
to this polytope