Polytope of Type {18,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,27}*972
if this polytope has a name.
Group : SmallGroup(972,106)
Rank : 3
Schlafli Type : {18,27}
Number of vertices, edges, etc : 18, 243, 27
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {18,27,2} of size 1944
Vertex Figure Of :
   {2,18,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,9}*324, {6,27}*324
   9-fold quotients : {2,27}*108, {6,9}*108
   27-fold quotients : {2,9}*36, {6,3}*36
   81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {18,54}*1944b
Permutation Representation (GAP) :

s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 58)( 29, 59)( 30, 60)( 31, 55)( 32, 56)( 33, 57)( 34, 61)
( 35, 62)( 36, 63)( 37, 67)( 38, 68)( 39, 69)( 40, 64)( 41, 65)( 42, 66)
( 43, 70)( 44, 71)( 45, 72)( 46, 76)( 47, 77)( 48, 78)( 49, 73)( 50, 74)
( 51, 75)( 52, 79)( 53, 80)( 54, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,139)(110,140)(111,141)
(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)(119,149)
(120,150)(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)(127,157)
(128,158)(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,220)(191,221)(192,222)(193,217)(194,218)(195,219)(196,223)
(197,224)(198,225)(199,229)(200,230)(201,231)(202,226)(203,227)(204,228)
(205,232)(206,233)(207,234)(208,238)(209,239)(210,240)(211,235)(212,236)
(213,237)(214,241)(215,242)(216,243);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)( 16, 51)
( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 41)( 27, 40)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 78)
( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)( 72, 79)
( 82,210)( 83,209)( 84,208)( 85,216)( 86,215)( 87,214)( 88,213)( 89,212)
( 90,211)( 91,201)( 92,200)( 93,199)( 94,207)( 95,206)( 96,205)( 97,204)
( 98,203)( 99,202)(100,192)(101,191)(102,190)(103,198)(104,197)(105,196)
(106,195)(107,194)(108,193)(109,183)(110,182)(111,181)(112,189)(113,188)
(114,187)(115,186)(116,185)(117,184)(118,174)(119,173)(120,172)(121,180)
(122,179)(123,178)(124,177)(125,176)(126,175)(127,165)(128,164)(129,163)
(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,240)(137,239)
(138,238)(139,237)(140,236)(141,235)(142,243)(143,242)(144,241)(145,231)
(146,230)(147,229)(148,228)(149,227)(150,226)(151,234)(152,233)(153,232)
(154,222)(155,221)(156,220)(157,219)(158,218)(159,217)(160,225)(161,224)
(162,223);;
s2 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10,102)( 11,101)( 12,100)( 13,108)( 14,107)( 15,106)( 16,105)
( 17,104)( 18,103)( 19, 93)( 20, 92)( 21, 91)( 22, 99)( 23, 98)( 24, 97)
( 25, 96)( 26, 95)( 27, 94)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)
( 33,137)( 34,142)( 35,144)( 36,143)( 37,159)( 38,158)( 39,157)( 40,156)
( 41,155)( 42,154)( 43,162)( 44,161)( 45,160)( 46,150)( 47,149)( 48,148)
( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,112)( 56,114)
( 57,113)( 58,109)( 59,111)( 60,110)( 61,115)( 62,117)( 63,116)( 64,132)
( 65,131)( 66,130)( 67,129)( 68,128)( 69,127)( 70,135)( 71,134)( 72,133)
( 73,123)( 74,122)( 75,121)( 76,120)( 77,119)( 78,118)( 79,126)( 80,125)
( 81,124)(163,183)(164,182)(165,181)(166,189)(167,188)(168,187)(169,186)
(170,185)(171,184)(172,174)(175,180)(176,179)(177,178)(190,240)(191,239)
(192,238)(193,237)(194,236)(195,235)(196,243)(197,242)(198,241)(199,231)
(200,230)(201,229)(202,228)(203,227)(204,226)(205,234)(206,233)(207,232)
(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,225)(215,224)
(216,223);;
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*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(243)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 58)( 29, 59)( 30, 60)( 31, 55)( 32, 56)( 33, 57)
( 34, 61)( 35, 62)( 36, 63)( 37, 67)( 38, 68)( 39, 69)( 40, 64)( 41, 65)
( 42, 66)( 43, 70)( 44, 71)( 45, 72)( 46, 76)( 47, 77)( 48, 78)( 49, 73)
( 50, 74)( 51, 75)( 52, 79)( 53, 80)( 54, 81)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,139)(110,140)
(111,141)(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)
(119,149)(120,150)(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)
(127,157)(128,158)(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)
(135,162)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,220)(191,221)(192,222)(193,217)(194,218)(195,219)
(196,223)(197,224)(198,225)(199,229)(200,230)(201,231)(202,226)(203,227)
(204,228)(205,232)(206,233)(207,234)(208,238)(209,239)(210,240)(211,235)
(212,236)(213,237)(214,241)(215,242)(216,243);
s1 := Sym(243)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 48)( 11, 47)( 12, 46)( 13, 54)( 14, 53)( 15, 52)
( 16, 51)( 17, 50)( 18, 49)( 19, 39)( 20, 38)( 21, 37)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 41)( 27, 40)( 55, 58)( 56, 60)( 57, 59)( 62, 63)
( 64, 78)( 65, 77)( 66, 76)( 67, 75)( 68, 74)( 69, 73)( 70, 81)( 71, 80)
( 72, 79)( 82,210)( 83,209)( 84,208)( 85,216)( 86,215)( 87,214)( 88,213)
( 89,212)( 90,211)( 91,201)( 92,200)( 93,199)( 94,207)( 95,206)( 96,205)
( 97,204)( 98,203)( 99,202)(100,192)(101,191)(102,190)(103,198)(104,197)
(105,196)(106,195)(107,194)(108,193)(109,183)(110,182)(111,181)(112,189)
(113,188)(114,187)(115,186)(116,185)(117,184)(118,174)(119,173)(120,172)
(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,165)(128,164)
(129,163)(130,171)(131,170)(132,169)(133,168)(134,167)(135,166)(136,240)
(137,239)(138,238)(139,237)(140,236)(141,235)(142,243)(143,242)(144,241)
(145,231)(146,230)(147,229)(148,228)(149,227)(150,226)(151,234)(152,233)
(153,232)(154,222)(155,221)(156,220)(157,219)(158,218)(159,217)(160,225)
(161,224)(162,223);
s2 := Sym(243)!(  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)
(  8, 87)(  9, 86)( 10,102)( 11,101)( 12,100)( 13,108)( 14,107)( 15,106)
( 16,105)( 17,104)( 18,103)( 19, 93)( 20, 92)( 21, 91)( 22, 99)( 23, 98)
( 24, 97)( 25, 96)( 26, 95)( 27, 94)( 28,139)( 29,141)( 30,140)( 31,136)
( 32,138)( 33,137)( 34,142)( 35,144)( 36,143)( 37,159)( 38,158)( 39,157)
( 40,156)( 41,155)( 42,154)( 43,162)( 44,161)( 45,160)( 46,150)( 47,149)
( 48,148)( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,112)
( 56,114)( 57,113)( 58,109)( 59,111)( 60,110)( 61,115)( 62,117)( 63,116)
( 64,132)( 65,131)( 66,130)( 67,129)( 68,128)( 69,127)( 70,135)( 71,134)
( 72,133)( 73,123)( 74,122)( 75,121)( 76,120)( 77,119)( 78,118)( 79,126)
( 80,125)( 81,124)(163,183)(164,182)(165,181)(166,189)(167,188)(168,187)
(169,186)(170,185)(171,184)(172,174)(175,180)(176,179)(177,178)(190,240)
(191,239)(192,238)(193,237)(194,236)(195,235)(196,243)(197,242)(198,241)
(199,231)(200,230)(201,229)(202,228)(203,227)(204,226)(205,234)(206,233)
(207,232)(208,222)(209,221)(210,220)(211,219)(212,218)(213,217)(214,225)
(215,224)(216,223);
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*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope