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# Polytope of Type {18,9}

Atlas Canonical Name : {18,9}*972i
if this polytope has a name.
Group : SmallGroup(972,110)
Rank : 3
Schlafli Type : {18,9}
Number of vertices, edges, etc : 54, 243, 27
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,9,2} of size 1944
Vertex Figure Of :
{2,18,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,9}*324a, {18,3}*324
9-fold quotients : {6,9}*108, {6,3}*108
27-fold quotients : {2,9}*36, {6,3}*36
81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944aa
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, 56)( 29, 55)( 30, 57)( 31, 62)
( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)
( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)
( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)(113,142)
(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)(121,152)
(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)(129,156)
(130,161)(131,160)(132,162)(133,158)(134,157)(135,159)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,218)(191,217)(192,219)(193,224)(194,223)(195,225)
(196,221)(197,220)(198,222)(199,227)(200,226)(201,228)(202,233)(203,232)
(204,234)(205,230)(206,229)(207,231)(208,236)(209,235)(210,237)(211,242)
(212,241)(213,243)(214,239)(215,238)(216,240);;
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, 56)( 59, 60)( 61, 63)( 64, 74)( 65, 73)
( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)( 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)(136,236)(137,235)(138,237)
(139,238)(140,240)(141,239)(142,243)(143,242)(144,241)(145,227)(146,226)
(147,228)(148,229)(149,231)(150,230)(151,234)(152,233)(153,232)(154,218)
(155,217)(156,219)(157,220)(158,222)(159,221)(160,225)(161,224)(162,223);;
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)( 28,143)( 29,142)( 30,144)( 31,137)( 32,136)
( 33,138)( 34,140)( 35,139)( 36,141)( 37,161)( 38,160)( 39,162)( 40,155)
( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,152)( 47,151)( 48,153)
( 49,146)( 50,145)( 51,147)( 52,149)( 53,148)( 54,150)( 55,113)( 56,112)
( 57,114)( 58,116)( 59,115)( 60,117)( 61,110)( 62,109)( 63,111)( 64,131)
( 65,130)( 66,132)( 67,134)( 68,133)( 69,135)( 70,128)( 71,127)( 72,129)
( 73,122)( 74,121)( 75,123)( 76,125)( 77,124)( 78,126)( 79,119)( 80,118)
( 81,120)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)(169,187)
(170,189)(171,188)(173,174)(176,177)(179,180)(190,242)(191,241)(192,243)
(193,236)(194,235)(195,237)(196,239)(197,238)(198,240)(199,233)(200,232)
(201,234)(202,227)(203,226)(204,228)(205,230)(206,229)(207,231)(208,224)
(209,223)(210,225)(211,218)(212,217)(213,219)(214,221)(215,220)(216,222);;
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*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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 ];;
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, 56)( 29, 55)( 30, 57)
( 31, 62)( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)
( 39, 66)( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)
( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)
(113,142)(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)
(121,152)(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)
(129,156)(130,161)(131,160)(132,162)(133,158)(134,157)(135,159)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(190,218)(191,217)(192,219)(193,224)(194,223)
(195,225)(196,221)(197,220)(198,222)(199,227)(200,226)(201,228)(202,233)
(203,232)(204,234)(205,230)(206,229)(207,231)(208,236)(209,235)(210,237)
(211,242)(212,241)(213,243)(214,239)(215,238)(216,240);
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, 56)( 59, 60)( 61, 63)( 64, 74)
( 65, 73)( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)
( 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)(136,236)(137,235)
(138,237)(139,238)(140,240)(141,239)(142,243)(143,242)(144,241)(145,227)
(146,226)(147,228)(148,229)(149,231)(150,230)(151,234)(152,233)(153,232)
(154,218)(155,217)(156,219)(157,220)(158,222)(159,221)(160,225)(161,224)
(162,223);
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)( 28,143)( 29,142)( 30,144)( 31,137)
( 32,136)( 33,138)( 34,140)( 35,139)( 36,141)( 37,161)( 38,160)( 39,162)
( 40,155)( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,152)( 47,151)
( 48,153)( 49,146)( 50,145)( 51,147)( 52,149)( 53,148)( 54,150)( 55,113)
( 56,112)( 57,114)( 58,116)( 59,115)( 60,117)( 61,110)( 62,109)( 63,111)
( 64,131)( 65,130)( 66,132)( 67,134)( 68,133)( 69,135)( 70,128)( 71,127)
( 72,129)( 73,122)( 74,121)( 75,123)( 76,125)( 77,124)( 78,126)( 79,119)
( 80,118)( 81,120)(163,181)(164,183)(165,182)(166,184)(167,186)(168,185)
(169,187)(170,189)(171,188)(173,174)(176,177)(179,180)(190,242)(191,241)
(192,243)(193,236)(194,235)(195,237)(196,239)(197,238)(198,240)(199,233)
(200,232)(201,234)(202,227)(203,226)(204,228)(205,230)(206,229)(207,231)
(208,224)(209,223)(210,225)(211,218)(212,217)(213,219)(214,221)(215,220)
(216,222);
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*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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 >;

References : None.
to this polytope