Polytope of Type {12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2322)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 54, 486, 81
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie
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}*216c
27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 5)( 7, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)
( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 28, 30)( 32, 33)( 34, 35)( 37, 48)
( 38, 47)( 39, 46)( 40, 49)( 41, 51)( 42, 50)( 43, 53)( 44, 52)( 45, 54)
( 55, 56)( 58, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 78)( 68, 77)
( 69, 76)( 70, 79)( 71, 81)( 72, 80)( 82,166)( 83,168)( 84,167)( 85,170)
( 86,169)( 87,171)( 88,165)( 89,164)( 90,163)( 91,184)( 92,186)( 93,185)
( 94,188)( 95,187)( 96,189)( 97,183)( 98,182)( 99,181)(100,175)(101,177)
(102,176)(103,179)(104,178)(105,180)(106,174)(107,173)(108,172)(109,195)
(110,194)(111,193)(112,196)(113,198)(114,197)(115,191)(116,190)(117,192)
(118,213)(119,212)(120,211)(121,214)(122,216)(123,215)(124,209)(125,208)
(126,210)(127,204)(128,203)(129,202)(130,205)(131,207)(132,206)(133,200)
(134,199)(135,201)(136,221)(137,220)(138,222)(139,225)(140,224)(141,223)
(142,217)(143,219)(144,218)(145,239)(146,238)(147,240)(148,243)(149,242)
(150,241)(151,235)(152,237)(153,236)(154,230)(155,229)(156,231)(157,234)
(158,233)(159,232)(160,226)(161,228)(162,227);;
s1 := ( 1, 10)( 2, 17)( 3, 15)( 4, 16)( 5, 14)( 6, 12)( 7, 13)( 8, 11)
( 9, 18)( 20, 26)( 21, 24)( 22, 25)( 28, 97)( 29, 95)( 30, 93)( 31, 94)
( 32, 92)( 33, 99)( 34, 91)( 35, 98)( 36, 96)( 37, 88)( 38, 86)( 39, 84)
( 40, 85)( 41, 83)( 42, 90)( 43, 82)( 44, 89)( 45, 87)( 46,106)( 47,104)
( 48,102)( 49,103)( 50,101)( 51,108)( 52,100)( 53,107)( 54,105)( 55,175)
( 56,173)( 57,180)( 58,172)( 59,179)( 60,177)( 61,178)( 62,176)( 63,174)
( 64,166)( 65,164)( 66,171)( 67,163)( 68,170)( 69,168)( 70,169)( 71,167)
( 72,165)( 73,184)( 74,182)( 75,189)( 76,181)( 77,188)( 78,186)( 79,187)
( 80,185)( 81,183)(109,121)(110,119)(111,126)(112,118)(113,125)(114,123)
(115,124)(116,122)(117,120)(127,130)(129,135)(131,134)(136,199)(137,206)
(138,204)(139,205)(140,203)(141,201)(142,202)(143,200)(144,207)(145,190)
(146,197)(147,195)(148,196)(149,194)(150,192)(151,193)(152,191)(153,198)
(154,208)(155,215)(156,213)(157,214)(158,212)(159,210)(160,211)(161,209)
(162,216)(217,232)(218,230)(219,228)(220,229)(221,227)(222,234)(223,226)
(224,233)(225,231)(235,241)(236,239)(240,243);;
s2 := ( 1, 31)( 2, 33)( 3, 32)( 4, 28)( 5, 30)( 6, 29)( 7, 34)( 8, 36)
( 9, 35)( 10, 40)( 11, 42)( 12, 41)( 13, 37)( 14, 39)( 15, 38)( 16, 43)
( 17, 45)( 18, 44)( 19, 49)( 20, 51)( 21, 50)( 22, 46)( 23, 48)( 24, 47)
( 25, 52)( 26, 54)( 27, 53)( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 65)
( 67, 71)( 68, 70)( 69, 72)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82,194)
( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)( 88,197)( 89,196)( 90,198)
( 91,203)( 92,202)( 93,204)( 94,200)( 95,199)( 96,201)( 97,206)( 98,205)
( 99,207)(100,212)(101,211)(102,213)(103,209)(104,208)(105,210)(106,215)
(107,214)(108,216)(109,167)(110,166)(111,168)(112,164)(113,163)(114,165)
(115,170)(116,169)(117,171)(118,176)(119,175)(120,177)(121,173)(122,172)
(123,174)(124,179)(125,178)(126,180)(127,185)(128,184)(129,186)(130,182)
(131,181)(132,183)(133,188)(134,187)(135,189)(136,219)(137,218)(138,217)
(139,225)(140,224)(141,223)(142,222)(143,221)(144,220)(145,228)(146,227)
(147,226)(148,234)(149,233)(150,232)(151,231)(152,230)(153,229)(154,237)
(155,236)(156,235)(157,243)(158,242)(159,241)(160,240)(161,239)(162,238);;
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*s0*s1*s2*s0*s1*s2*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*s0*s1*s2*s1*s2*s1*s0*s1*s0*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*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 2, 3)( 4, 5)( 7, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)
( 14, 22)( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 28, 30)( 32, 33)( 34, 35)
( 37, 48)( 38, 47)( 39, 46)( 40, 49)( 41, 51)( 42, 50)( 43, 53)( 44, 52)
( 45, 54)( 55, 56)( 58, 60)( 62, 63)( 64, 74)( 65, 73)( 66, 75)( 67, 78)
( 68, 77)( 69, 76)( 70, 79)( 71, 81)( 72, 80)( 82,166)( 83,168)( 84,167)
( 85,170)( 86,169)( 87,171)( 88,165)( 89,164)( 90,163)( 91,184)( 92,186)
( 93,185)( 94,188)( 95,187)( 96,189)( 97,183)( 98,182)( 99,181)(100,175)
(101,177)(102,176)(103,179)(104,178)(105,180)(106,174)(107,173)(108,172)
(109,195)(110,194)(111,193)(112,196)(113,198)(114,197)(115,191)(116,190)
(117,192)(118,213)(119,212)(120,211)(121,214)(122,216)(123,215)(124,209)
(125,208)(126,210)(127,204)(128,203)(129,202)(130,205)(131,207)(132,206)
(133,200)(134,199)(135,201)(136,221)(137,220)(138,222)(139,225)(140,224)
(141,223)(142,217)(143,219)(144,218)(145,239)(146,238)(147,240)(148,243)
(149,242)(150,241)(151,235)(152,237)(153,236)(154,230)(155,229)(156,231)
(157,234)(158,233)(159,232)(160,226)(161,228)(162,227);
s1 := Sym(243)!( 1, 10)( 2, 17)( 3, 15)( 4, 16)( 5, 14)( 6, 12)( 7, 13)
( 8, 11)( 9, 18)( 20, 26)( 21, 24)( 22, 25)( 28, 97)( 29, 95)( 30, 93)
( 31, 94)( 32, 92)( 33, 99)( 34, 91)( 35, 98)( 36, 96)( 37, 88)( 38, 86)
( 39, 84)( 40, 85)( 41, 83)( 42, 90)( 43, 82)( 44, 89)( 45, 87)( 46,106)
( 47,104)( 48,102)( 49,103)( 50,101)( 51,108)( 52,100)( 53,107)( 54,105)
( 55,175)( 56,173)( 57,180)( 58,172)( 59,179)( 60,177)( 61,178)( 62,176)
( 63,174)( 64,166)( 65,164)( 66,171)( 67,163)( 68,170)( 69,168)( 70,169)
( 71,167)( 72,165)( 73,184)( 74,182)( 75,189)( 76,181)( 77,188)( 78,186)
( 79,187)( 80,185)( 81,183)(109,121)(110,119)(111,126)(112,118)(113,125)
(114,123)(115,124)(116,122)(117,120)(127,130)(129,135)(131,134)(136,199)
(137,206)(138,204)(139,205)(140,203)(141,201)(142,202)(143,200)(144,207)
(145,190)(146,197)(147,195)(148,196)(149,194)(150,192)(151,193)(152,191)
(153,198)(154,208)(155,215)(156,213)(157,214)(158,212)(159,210)(160,211)
(161,209)(162,216)(217,232)(218,230)(219,228)(220,229)(221,227)(222,234)
(223,226)(224,233)(225,231)(235,241)(236,239)(240,243);
s2 := Sym(243)!( 1, 31)( 2, 33)( 3, 32)( 4, 28)( 5, 30)( 6, 29)( 7, 34)
( 8, 36)( 9, 35)( 10, 40)( 11, 42)( 12, 41)( 13, 37)( 14, 39)( 15, 38)
( 16, 43)( 17, 45)( 18, 44)( 19, 49)( 20, 51)( 21, 50)( 22, 46)( 23, 48)
( 24, 47)( 25, 52)( 26, 54)( 27, 53)( 55, 56)( 58, 62)( 59, 61)( 60, 63)
( 64, 65)( 67, 71)( 68, 70)( 69, 72)( 73, 74)( 76, 80)( 77, 79)( 78, 81)
( 82,194)( 83,193)( 84,195)( 85,191)( 86,190)( 87,192)( 88,197)( 89,196)
( 90,198)( 91,203)( 92,202)( 93,204)( 94,200)( 95,199)( 96,201)( 97,206)
( 98,205)( 99,207)(100,212)(101,211)(102,213)(103,209)(104,208)(105,210)
(106,215)(107,214)(108,216)(109,167)(110,166)(111,168)(112,164)(113,163)
(114,165)(115,170)(116,169)(117,171)(118,176)(119,175)(120,177)(121,173)
(122,172)(123,174)(124,179)(125,178)(126,180)(127,185)(128,184)(129,186)
(130,182)(131,181)(132,183)(133,188)(134,187)(135,189)(136,219)(137,218)
(138,217)(139,225)(140,224)(141,223)(142,222)(143,221)(144,220)(145,228)
(146,227)(147,226)(148,234)(149,233)(150,232)(151,231)(152,230)(153,229)
(154,237)(155,236)(156,235)(157,243)(158,242)(159,241)(160,240)(161,239)
(162,238);
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*s0*s1*s2*s0*s1*s2*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*s0*s1*s2*s1*s2*s1*s0*s1*s0*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*s2 >;
References : None.
to this polytope