Polytope of Type {4,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18}*1944a
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {4,18}
Number of vertices, edges, etc : 54, 486, 243
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,18}*648
   9-fold quotients : {4,6}*216
   27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope