Polytope of Type {36,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,12}*864b
if this polytope has a 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 : 6
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}*432b, {18,12}*432b
   3-fold quotients : {36,4}*288a, {12,12}*288c
   4-fold quotients : {18,6}*216b
   6-fold quotients : {36,2}*144, {18,4}*144a, {12,6}*144b, {6,12}*144c
   8-fold quotients : {9,6}*108
   9-fold quotients : {12,4}*96a
   12-fold quotients : {18,2}*72, {6,6}*72c
   18-fold quotients : {12,2}*48, {6,4}*48a
   24-fold quotients : {9,2}*36, {3,6}*36
   27-fold quotients : {4,4}*32
   36-fold quotients : {6,2}*24
   54-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,24}*1728a, {36,12}*1728b, {36,24}*1728b, {72,12}*1728b, {72,12}*1728d
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope