Polytope of Type {9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,18}*972j
if this polytope has a name.
Group : SmallGroup(972,111)
Rank : 3
Schlafli Type : {9,18}
Number of vertices, edges, etc : 27, 243, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{9,18,2} of size 1944
Vertex Figure Of :
{2,9,18} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {9,6}*324b, {3,18}*324
9-fold quotients : {3,6}*108
27-fold quotients : {3,6}*36
81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,18}*1944ab
Permutation Representation (GAP) :
s0 := ( 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, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)
( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)
( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 82,181)
( 83,183)( 84,182)( 85,186)( 86,185)( 87,184)( 88,188)( 89,187)( 90,189)
( 91,172)( 92,174)( 93,173)( 94,177)( 95,176)( 96,175)( 97,179)( 98,178)
( 99,180)(100,163)(101,165)(102,164)(103,168)(104,167)(105,166)(106,170)
(107,169)(108,171)(109,235)(110,237)(111,236)(112,240)(113,239)(114,238)
(115,242)(116,241)(117,243)(118,226)(119,228)(120,227)(121,231)(122,230)
(123,229)(124,233)(125,232)(126,234)(127,217)(128,219)(129,218)(130,222)
(131,221)(132,220)(133,224)(134,223)(135,225)(136,208)(137,210)(138,209)
(139,213)(140,212)(141,211)(142,215)(143,214)(144,216)(145,199)(146,201)
(147,200)(148,204)(149,203)(150,202)(151,206)(152,205)(153,207)(154,190)
(155,192)(156,191)(157,195)(158,194)(159,193)(160,197)(161,196)(162,198);;
s1 := ( 1,193)( 2,195)( 3,194)( 4,196)( 5,198)( 6,197)( 7,190)( 8,192)
( 9,191)( 10,211)( 11,213)( 12,212)( 13,214)( 14,216)( 15,215)( 16,208)
( 17,210)( 18,209)( 19,202)( 20,204)( 21,203)( 22,205)( 23,207)( 24,206)
( 25,199)( 26,201)( 27,200)( 28,169)( 29,171)( 30,170)( 31,163)( 32,165)
( 33,164)( 34,166)( 35,168)( 36,167)( 37,187)( 38,189)( 39,188)( 40,181)
( 41,183)( 42,182)( 43,184)( 44,186)( 45,185)( 46,178)( 47,180)( 48,179)
( 49,172)( 50,174)( 51,173)( 52,175)( 53,177)( 54,176)( 55,219)( 56,218)
( 57,217)( 58,222)( 59,221)( 60,220)( 61,225)( 62,224)( 63,223)( 64,237)
( 65,236)( 66,235)( 67,240)( 68,239)( 69,238)( 70,243)( 71,242)( 72,241)
( 73,228)( 74,227)( 75,226)( 76,231)( 77,230)( 78,229)( 79,234)( 80,233)
( 81,232)( 82,112)( 83,114)( 84,113)( 85,115)( 86,117)( 87,116)( 88,109)
( 89,111)( 90,110)( 91,130)( 92,132)( 93,131)( 94,133)( 95,135)( 96,134)
( 97,127)( 98,129)( 99,128)(100,121)(101,123)(102,122)(103,124)(104,126)
(105,125)(106,118)(107,120)(108,119)(136,138)(139,141)(142,144)(145,156)
(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)(153,160);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)
( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 82,181)
( 83,183)( 84,182)( 85,187)( 86,189)( 87,188)( 88,184)( 89,186)( 90,185)
( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)( 98,177)
( 99,176)(100,163)(101,165)(102,164)(103,169)(104,171)(105,170)(106,166)
(107,168)(108,167)(109,208)(110,210)(111,209)(112,214)(113,216)(114,215)
(115,211)(116,213)(117,212)(118,199)(119,201)(120,200)(121,205)(122,207)
(123,206)(124,202)(125,204)(126,203)(127,190)(128,192)(129,191)(130,196)
(131,198)(132,197)(133,193)(134,195)(135,194)(136,235)(137,237)(138,236)
(139,241)(140,243)(141,242)(142,238)(143,240)(144,239)(145,226)(146,228)
(147,227)(148,232)(149,234)(150,233)(151,229)(152,231)(153,230)(154,217)
(155,219)(156,218)(157,223)(158,225)(159,224)(160,220)(161,222)(162,221);;
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*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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, 55)( 29, 57)( 30, 56)
( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)
( 39, 74)( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)
( 47, 66)( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)
( 82,181)( 83,183)( 84,182)( 85,186)( 86,185)( 87,184)( 88,188)( 89,187)
( 90,189)( 91,172)( 92,174)( 93,173)( 94,177)( 95,176)( 96,175)( 97,179)
( 98,178)( 99,180)(100,163)(101,165)(102,164)(103,168)(104,167)(105,166)
(106,170)(107,169)(108,171)(109,235)(110,237)(111,236)(112,240)(113,239)
(114,238)(115,242)(116,241)(117,243)(118,226)(119,228)(120,227)(121,231)
(122,230)(123,229)(124,233)(125,232)(126,234)(127,217)(128,219)(129,218)
(130,222)(131,221)(132,220)(133,224)(134,223)(135,225)(136,208)(137,210)
(138,209)(139,213)(140,212)(141,211)(142,215)(143,214)(144,216)(145,199)
(146,201)(147,200)(148,204)(149,203)(150,202)(151,206)(152,205)(153,207)
(154,190)(155,192)(156,191)(157,195)(158,194)(159,193)(160,197)(161,196)
(162,198);
s1 := Sym(243)!( 1,193)( 2,195)( 3,194)( 4,196)( 5,198)( 6,197)( 7,190)
( 8,192)( 9,191)( 10,211)( 11,213)( 12,212)( 13,214)( 14,216)( 15,215)
( 16,208)( 17,210)( 18,209)( 19,202)( 20,204)( 21,203)( 22,205)( 23,207)
( 24,206)( 25,199)( 26,201)( 27,200)( 28,169)( 29,171)( 30,170)( 31,163)
( 32,165)( 33,164)( 34,166)( 35,168)( 36,167)( 37,187)( 38,189)( 39,188)
( 40,181)( 41,183)( 42,182)( 43,184)( 44,186)( 45,185)( 46,178)( 47,180)
( 48,179)( 49,172)( 50,174)( 51,173)( 52,175)( 53,177)( 54,176)( 55,219)
( 56,218)( 57,217)( 58,222)( 59,221)( 60,220)( 61,225)( 62,224)( 63,223)
( 64,237)( 65,236)( 66,235)( 67,240)( 68,239)( 69,238)( 70,243)( 71,242)
( 72,241)( 73,228)( 74,227)( 75,226)( 76,231)( 77,230)( 78,229)( 79,234)
( 80,233)( 81,232)( 82,112)( 83,114)( 84,113)( 85,115)( 86,117)( 87,116)
( 88,109)( 89,111)( 90,110)( 91,130)( 92,132)( 93,131)( 94,133)( 95,135)
( 96,134)( 97,127)( 98,129)( 99,128)(100,121)(101,123)(102,122)(103,124)
(104,126)(105,125)(106,118)(107,120)(108,119)(136,138)(139,141)(142,144)
(145,156)(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)
(153,160);
s2 := Sym(243)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 19)( 11, 21)( 12, 20)
( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)
( 43, 49)( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)
( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)
( 82,181)( 83,183)( 84,182)( 85,187)( 86,189)( 87,188)( 88,184)( 89,186)
( 90,185)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)
( 98,177)( 99,176)(100,163)(101,165)(102,164)(103,169)(104,171)(105,170)
(106,166)(107,168)(108,167)(109,208)(110,210)(111,209)(112,214)(113,216)
(114,215)(115,211)(116,213)(117,212)(118,199)(119,201)(120,200)(121,205)
(122,207)(123,206)(124,202)(125,204)(126,203)(127,190)(128,192)(129,191)
(130,196)(131,198)(132,197)(133,193)(134,195)(135,194)(136,235)(137,237)
(138,236)(139,241)(140,243)(141,242)(142,238)(143,240)(144,239)(145,226)
(146,228)(147,227)(148,232)(149,234)(150,233)(151,229)(152,231)(153,230)
(154,217)(155,219)(156,218)(157,223)(158,225)(159,224)(160,220)(161,222)
(162,221);
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*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;
References : None.
to this polytope