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