Polytope of Type {36,12} with group SmallGroup(864,1018)

Overview

Group: SmallGroup(864,1018)
Rank: 3
Schläfli Type: {36,12}
Vertices, edges, …: 36, 216, 12
Order of s₀s₁s₂: 36
Order of s₀s₁s₂s₁: 2
Also known as: {36,12|2}. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

{18,6}*216a

6-fold

9-fold

12-fold

18-fold

24-fold

{9,2}*36

27-fold

{4,4}*32

36-fold

54-fold

72-fold

108-fold

{2,2}*8

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References