Polytope of Type {12,18} with group SmallGroup(1944,805)

Overview

Group: SmallGroup(1944,805)
Rank: 3
Schläfli Type: {12,18}
Vertices, edges, …: 54, 486, 81
Order of s₀s₁s₂: 12
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie

Quotients maximal quotients in bold

9-fold

{12,6}*216b

27-fold

{4,6}*72

Covers minimal covers in bold

None in this atlas.

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*s2)^5*s1*s0*s2> of order 3

27 facets

27 of {12}*24

36 vertex figures

9 of {18}*36
27 of {6}*12

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

27 facets

27 of {12}*24

18 vertex figures

18 of {18}*36

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.