Polytope of Type {36,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,12}*864a
Also Known As : {36,12|2}. if this polytope has another name.
Group : SmallGroup(864,1018)
Rank : 3
Schlafli Type : {36,12}
Number of vertices, edges, etc : 36, 216, 12
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {36,12,2} of size 1728
Vertex Figure Of :
   {2,36,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,6}*432a, {18,12}*432a
   3-fold quotients : {36,4}*288a, {12,12}*288a
   4-fold quotients : {18,6}*216a
   6-fold quotients : {36,2}*144, {18,4}*144a, {6,12}*144a, {12,6}*144a
   9-fold quotients : {4,12}*96a, {12,4}*96a
   12-fold quotients : {18,2}*72, {6,6}*72a
   18-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
   24-fold quotients : {9,2}*36
   27-fold quotients : {4,4}*32
   36-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {2,3}*12, {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,12}*1728a, {72,12}*1728a, {36,24}*1728c, {72,12}*1728c, {36,24}*1728d
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 : None.
to this polytope