Polytope of Type {18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,18}*972a
if this polytope has a name.
Group : SmallGroup(972,99)
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}*324a, {18,6}*324a
9-fold quotients : {6,6}*108
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944c
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, 61)( 29, 63)( 30, 62)( 31, 58)
( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)( 39, 71)
( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)( 47, 81)
( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)(113,141)
(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)(121,148)
(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)(129,161)
(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)(195,221)
(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)(203,231)
(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)(211,238)
(212,240)(213,239)(214,235)(215,237)(216,236);;
s1 := ( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)( 8, 32)
( 9, 33)( 10, 46)( 11, 47)( 12, 48)( 13, 52)( 14, 53)( 15, 54)( 16, 49)
( 17, 50)( 18, 51)( 19, 37)( 20, 38)( 21, 39)( 22, 43)( 23, 44)( 24, 45)
( 25, 40)( 26, 41)( 27, 42)( 55, 61)( 56, 62)( 57, 63)( 64, 79)( 65, 80)
( 66, 81)( 67, 76)( 68, 77)( 69, 78)( 70, 73)( 71, 74)( 72, 75)( 82,208)
( 83,209)( 84,210)( 85,214)( 86,215)( 87,216)( 88,211)( 89,212)( 90,213)
( 91,199)( 92,200)( 93,201)( 94,205)( 95,206)( 96,207)( 97,202)( 98,203)
( 99,204)(100,190)(101,191)(102,192)(103,196)(104,197)(105,198)(106,193)
(107,194)(108,195)(109,181)(110,182)(111,183)(112,187)(113,188)(114,189)
(115,184)(116,185)(117,186)(118,172)(119,173)(120,174)(121,178)(122,179)
(123,180)(124,175)(125,176)(126,177)(127,163)(128,164)(129,165)(130,169)
(131,170)(132,171)(133,166)(134,167)(135,168)(136,241)(137,242)(138,243)
(139,238)(140,239)(141,240)(142,235)(143,236)(144,237)(145,232)(146,233)
(147,234)(148,229)(149,230)(150,231)(151,226)(152,227)(153,228)(154,223)
(155,224)(156,225)(157,220)(158,221)(159,222)(160,217)(161,218)(162,219);;
s2 := ( 1, 82)( 2, 84)( 3, 83)( 4, 85)( 5, 87)( 6, 86)( 7, 88)( 8, 90)
( 9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)( 16,106)
( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)( 24, 95)
( 25, 97)( 26, 99)( 27, 98)( 28,110)( 29,109)( 30,111)( 31,113)( 32,112)
( 33,114)( 34,116)( 35,115)( 36,117)( 37,128)( 38,127)( 39,129)( 40,131)
( 41,130)( 42,132)( 43,134)( 44,133)( 45,135)( 46,119)( 47,118)( 48,120)
( 49,122)( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,138)( 56,137)
( 57,136)( 58,141)( 59,140)( 60,139)( 61,144)( 62,143)( 63,142)( 64,156)
( 65,155)( 66,154)( 67,159)( 68,158)( 69,157)( 70,162)( 71,161)( 72,160)
( 73,147)( 74,146)( 75,145)( 76,150)( 77,149)( 78,148)( 79,153)( 80,152)
( 81,151)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)(169,187)
(170,189)(171,188)(173,174)(176,177)(179,180)(190,209)(191,208)(192,210)
(193,212)(194,211)(195,213)(196,215)(197,214)(198,216)(199,200)(202,203)
(205,206)(217,237)(218,236)(219,235)(220,240)(221,239)(222,238)(223,243)
(224,242)(225,241)(226,228)(229,231)(232,234);;
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*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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, 61)( 29, 63)( 30, 62)
( 31, 58)( 32, 60)( 33, 59)( 34, 55)( 35, 57)( 36, 56)( 37, 70)( 38, 72)
( 39, 71)( 40, 67)( 41, 69)( 42, 68)( 43, 64)( 44, 66)( 45, 65)( 46, 79)
( 47, 81)( 48, 80)( 49, 76)( 50, 78)( 51, 77)( 52, 73)( 53, 75)( 54, 74)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,142)(110,144)(111,143)(112,139)
(113,141)(114,140)(115,136)(116,138)(117,137)(118,151)(119,153)(120,152)
(121,148)(122,150)(123,149)(124,145)(125,147)(126,146)(127,160)(128,162)
(129,161)(130,157)(131,159)(132,158)(133,154)(134,156)(135,155)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,223)(191,225)(192,224)(193,220)(194,222)
(195,221)(196,217)(197,219)(198,218)(199,232)(200,234)(201,233)(202,229)
(203,231)(204,230)(205,226)(206,228)(207,227)(208,241)(209,243)(210,242)
(211,238)(212,240)(213,239)(214,235)(215,237)(216,236);
s1 := Sym(243)!( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)
( 8, 32)( 9, 33)( 10, 46)( 11, 47)( 12, 48)( 13, 52)( 14, 53)( 15, 54)
( 16, 49)( 17, 50)( 18, 51)( 19, 37)( 20, 38)( 21, 39)( 22, 43)( 23, 44)
( 24, 45)( 25, 40)( 26, 41)( 27, 42)( 55, 61)( 56, 62)( 57, 63)( 64, 79)
( 65, 80)( 66, 81)( 67, 76)( 68, 77)( 69, 78)( 70, 73)( 71, 74)( 72, 75)
( 82,208)( 83,209)( 84,210)( 85,214)( 86,215)( 87,216)( 88,211)( 89,212)
( 90,213)( 91,199)( 92,200)( 93,201)( 94,205)( 95,206)( 96,207)( 97,202)
( 98,203)( 99,204)(100,190)(101,191)(102,192)(103,196)(104,197)(105,198)
(106,193)(107,194)(108,195)(109,181)(110,182)(111,183)(112,187)(113,188)
(114,189)(115,184)(116,185)(117,186)(118,172)(119,173)(120,174)(121,178)
(122,179)(123,180)(124,175)(125,176)(126,177)(127,163)(128,164)(129,165)
(130,169)(131,170)(132,171)(133,166)(134,167)(135,168)(136,241)(137,242)
(138,243)(139,238)(140,239)(141,240)(142,235)(143,236)(144,237)(145,232)
(146,233)(147,234)(148,229)(149,230)(150,231)(151,226)(152,227)(153,228)
(154,223)(155,224)(156,225)(157,220)(158,221)(159,222)(160,217)(161,218)
(162,219);
s2 := Sym(243)!( 1, 82)( 2, 84)( 3, 83)( 4, 85)( 5, 87)( 6, 86)( 7, 88)
( 8, 90)( 9, 89)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)
( 16,106)( 17,108)( 18,107)( 19, 91)( 20, 93)( 21, 92)( 22, 94)( 23, 96)
( 24, 95)( 25, 97)( 26, 99)( 27, 98)( 28,110)( 29,109)( 30,111)( 31,113)
( 32,112)( 33,114)( 34,116)( 35,115)( 36,117)( 37,128)( 38,127)( 39,129)
( 40,131)( 41,130)( 42,132)( 43,134)( 44,133)( 45,135)( 46,119)( 47,118)
( 48,120)( 49,122)( 50,121)( 51,123)( 52,125)( 53,124)( 54,126)( 55,138)
( 56,137)( 57,136)( 58,141)( 59,140)( 60,139)( 61,144)( 62,143)( 63,142)
( 64,156)( 65,155)( 66,154)( 67,159)( 68,158)( 69,157)( 70,162)( 71,161)
( 72,160)( 73,147)( 74,146)( 75,145)( 76,150)( 77,149)( 78,148)( 79,153)
( 80,152)( 81,151)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)
(169,187)(170,189)(171,188)(173,174)(176,177)(179,180)(190,209)(191,208)
(192,210)(193,212)(194,211)(195,213)(196,215)(197,214)(198,216)(199,200)
(202,203)(205,206)(217,237)(218,236)(219,235)(220,240)(221,239)(222,238)
(223,243)(224,242)(225,241)(226,228)(229,231)(232,234);
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*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 >;
References : None.
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