Polytope of Type {12,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1944b
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 54, 486, 81
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
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 : {12,6}*216a
   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 := (  1, 10)(  2, 12)(  3, 11)(  4, 15)(  5, 14)(  6, 13)(  7, 17)(  8, 16)
(  9, 18)( 20, 21)( 22, 24)( 25, 26)( 28,168)( 29,167)( 30,166)( 31,170)
( 32,169)( 33,171)( 34,163)( 35,165)( 36,164)( 37,186)( 38,185)( 39,184)
( 40,188)( 41,187)( 42,189)( 43,181)( 44,183)( 45,182)( 46,177)( 47,176)
( 48,175)( 49,179)( 50,178)( 51,180)( 52,172)( 53,174)( 54,173)( 55,101)
( 56,100)( 57,102)( 58,103)( 59,105)( 60,104)( 61,108)( 62,107)( 63,106)
( 64, 92)( 65, 91)( 66, 93)( 67, 94)( 68, 96)( 69, 95)( 70, 99)( 71, 98)
( 72, 97)( 73, 83)( 74, 82)( 75, 84)( 76, 85)( 77, 87)( 78, 86)( 79, 90)
( 80, 89)( 81, 88)(109,238)(110,240)(111,239)(112,243)(113,242)(114,241)
(115,236)(116,235)(117,237)(118,229)(119,231)(120,230)(121,234)(122,233)
(123,232)(124,227)(125,226)(126,228)(127,220)(128,222)(129,221)(130,225)
(131,224)(132,223)(133,218)(134,217)(135,219)(136,156)(137,155)(138,154)
(139,158)(140,157)(141,159)(142,160)(143,162)(144,161)(145,147)(148,149)
(152,153)(190,201)(191,200)(192,199)(193,203)(194,202)(195,204)(196,205)
(197,207)(198,206)(208,210)(211,212)(215,216);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2 ];;
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)!(  1, 10)(  2, 12)(  3, 11)(  4, 15)(  5, 14)(  6, 13)(  7, 17)
(  8, 16)(  9, 18)( 20, 21)( 22, 24)( 25, 26)( 28,168)( 29,167)( 30,166)
( 31,170)( 32,169)( 33,171)( 34,163)( 35,165)( 36,164)( 37,186)( 38,185)
( 39,184)( 40,188)( 41,187)( 42,189)( 43,181)( 44,183)( 45,182)( 46,177)
( 47,176)( 48,175)( 49,179)( 50,178)( 51,180)( 52,172)( 53,174)( 54,173)
( 55,101)( 56,100)( 57,102)( 58,103)( 59,105)( 60,104)( 61,108)( 62,107)
( 63,106)( 64, 92)( 65, 91)( 66, 93)( 67, 94)( 68, 96)( 69, 95)( 70, 99)
( 71, 98)( 72, 97)( 73, 83)( 74, 82)( 75, 84)( 76, 85)( 77, 87)( 78, 86)
( 79, 90)( 80, 89)( 81, 88)(109,238)(110,240)(111,239)(112,243)(113,242)
(114,241)(115,236)(116,235)(117,237)(118,229)(119,231)(120,230)(121,234)
(122,233)(123,232)(124,227)(125,226)(126,228)(127,220)(128,222)(129,221)
(130,225)(131,224)(132,223)(133,218)(134,217)(135,219)(136,156)(137,155)
(138,154)(139,158)(140,157)(141,159)(142,160)(143,162)(144,161)(145,147)
(148,149)(152,153)(190,201)(191,200)(192,199)(193,203)(194,202)(195,204)
(196,205)(197,207)(198,206)(208,210)(211,212)(215,216);
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2 >;

References : None.
to this polytope