Polytope of Type {9,18} with group SmallGroup(972,111)

Overview

Group: SmallGroup(972,111)
Rank: 3
Schläfli Type: {9,18}
Vertices, edges, …: 27, 243, 54
Order of s₀s₁s₂: 6
Order of s₀s₁s₂s₁: 18
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

3-fold

9-fold

{3,6}*108

27-fold

{3,6}*36

81-fold

{3,2}*12

Covers minimal covers in bold

2-fold

{18,18}*1944ab

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 3

18 facets

18 of {9}*18

9 vertex figures

9 of {18}*36

P/N, where N=<s0*s1*s0*s2*s1*s0*(s1*s2)^3> of order 3

18 facets

18 of {9}*18

9 vertex figures

9 of {18}*36

Representations

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, 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*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, 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*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.