Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*972b
if this polytope has a name.
Group : SmallGroup(972,100)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 27, 243, 27
Order of s0s1s2 : 3
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}*324c, {18,6}*324c
9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944d
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)( 82,196)
( 83,198)( 84,197)( 85,193)( 86,195)( 87,194)( 88,190)( 89,192)( 90,191)
( 91,207)( 92,206)( 93,205)( 94,204)( 95,203)( 96,202)( 97,201)( 98,200)
( 99,199)(100,215)(101,214)(102,216)(103,212)(104,211)(105,213)(106,209)
(107,208)(108,210)(109,169)(110,171)(111,170)(112,166)(113,168)(114,167)
(115,163)(116,165)(117,164)(118,180)(119,179)(120,178)(121,177)(122,176)
(123,175)(124,174)(125,173)(126,172)(127,188)(128,187)(129,189)(130,185)
(131,184)(132,186)(133,182)(134,181)(135,183)(136,224)(137,223)(138,225)
(139,221)(140,220)(141,222)(142,218)(143,217)(144,219)(145,232)(146,234)
(147,233)(148,229)(149,231)(150,230)(151,226)(152,228)(153,227)(154,243)
(155,242)(156,241)(157,240)(158,239)(159,238)(160,237)(161,236)(162,235);;
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)(163,169)(164,170)(165,171)(172,187)(173,188)(174,189)(175,184)
(176,185)(177,186)(178,181)(179,182)(180,183)(190,205)(191,206)(192,207)
(193,202)(194,203)(195,204)(196,199)(197,200)(198,201)(208,214)(209,215)
(210,216)(217,235)(218,236)(219,237)(220,241)(221,242)(222,243)(223,238)
(224,239)(225,240)(229,232)(230,233)(231,234);;
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, s0*s1*s2*s0*s1*s2*s0*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,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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)
( 82,196)( 83,198)( 84,197)( 85,193)( 86,195)( 87,194)( 88,190)( 89,192)
( 90,191)( 91,207)( 92,206)( 93,205)( 94,204)( 95,203)( 96,202)( 97,201)
( 98,200)( 99,199)(100,215)(101,214)(102,216)(103,212)(104,211)(105,213)
(106,209)(107,208)(108,210)(109,169)(110,171)(111,170)(112,166)(113,168)
(114,167)(115,163)(116,165)(117,164)(118,180)(119,179)(120,178)(121,177)
(122,176)(123,175)(124,174)(125,173)(126,172)(127,188)(128,187)(129,189)
(130,185)(131,184)(132,186)(133,182)(134,181)(135,183)(136,224)(137,223)
(138,225)(139,221)(140,220)(141,222)(142,218)(143,217)(144,219)(145,232)
(146,234)(147,233)(148,229)(149,231)(150,230)(151,226)(152,228)(153,227)
(154,243)(155,242)(156,241)(157,240)(158,239)(159,238)(160,237)(161,236)
(162,235);
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)(163,169)(164,170)(165,171)(172,187)(173,188)(174,189)
(175,184)(176,185)(177,186)(178,181)(179,182)(180,183)(190,205)(191,206)
(192,207)(193,202)(194,203)(195,204)(196,199)(197,200)(198,201)(208,214)
(209,215)(210,216)(217,235)(218,236)(219,237)(220,241)(221,242)(222,243)
(223,238)(224,239)(225,240)(229,232)(230,233)(231,234);
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, s0*s1*s2*s0*s1*s2*s0*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,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope