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Polytope of Type {36,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,4,6}*1728
Also Known As : {{36,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(1728,14461)
Rank : 4
Schlafli Type : {36,4,6}
Number of vertices, edges, etc : 36, 72, 12, 6
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,2,6}*864, {18,4,6}*864
   3-fold quotients : {36,4,2}*576a, {12,4,6}*576
   4-fold quotients : {36,2,3}*432, {18,2,6}*432
   6-fold quotients : {36,2,2}*288, {18,4,2}*288a, {12,2,6}*288, {6,4,6}*288
   8-fold quotients : {9,2,6}*216, {18,2,3}*216
   9-fold quotients : {12,4,2}*192a, {4,4,6}*192
   12-fold quotients : {18,2,2}*144, {12,2,3}*144, {6,2,6}*144
   16-fold quotients : {9,2,3}*108
   18-fold quotients : {12,2,2}*96, {2,4,6}*96a, {4,2,6}*96, {6,4,2}*96a
   24-fold quotients : {9,2,2}*72, {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)
( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)( 33, 34)
( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)( 51, 52)
( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 65, 66)( 67, 72)( 68, 71)( 69, 70)
( 74, 75)( 76, 81)( 77, 80)( 78, 79)( 83, 84)( 85, 90)( 86, 89)( 87, 88)
( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)(105,106)
(109,136)(110,138)(111,137)(112,144)(113,143)(114,142)(115,141)(116,140)
(117,139)(118,145)(119,147)(120,146)(121,153)(122,152)(123,151)(124,150)
(125,149)(126,148)(127,154)(128,156)(129,155)(130,162)(131,161)(132,160)
(133,159)(134,158)(135,157)(163,190)(164,192)(165,191)(166,198)(167,197)
(168,196)(169,195)(170,194)(171,193)(172,199)(173,201)(174,200)(175,207)
(176,206)(177,205)(178,204)(179,203)(180,202)(181,208)(182,210)(183,209)
(184,216)(185,215)(186,214)(187,213)(188,212)(189,211);;
s1 := (  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,117)(  8,116)
(  9,115)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)( 16,126)
( 17,125)( 18,124)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)( 24,128)
( 25,135)( 26,134)( 27,133)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)
( 33,137)( 34,144)( 35,143)( 36,142)( 37,148)( 38,150)( 39,149)( 40,145)
( 41,147)( 42,146)( 43,153)( 44,152)( 45,151)( 46,157)( 47,159)( 48,158)
( 49,154)( 50,156)( 51,155)( 52,162)( 53,161)( 54,160)( 55,166)( 56,168)
( 57,167)( 58,163)( 59,165)( 60,164)( 61,171)( 62,170)( 63,169)( 64,175)
( 65,177)( 66,176)( 67,172)( 68,174)( 69,173)( 70,180)( 71,179)( 72,178)
( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)( 79,189)( 80,188)
( 81,187)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)( 88,198)
( 89,197)( 90,196)( 91,202)( 92,204)( 93,203)( 94,199)( 95,201)( 96,200)
( 97,207)( 98,206)( 99,205)(100,211)(101,213)(102,212)(103,208)(104,210)
(105,209)(106,216)(107,215)(108,214);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(109,163)(110,164)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)(119,182)(120,183)
(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,172)(128,173)
(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)
(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)
(161,206)(162,207);;
s3 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)(112,121)
(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)(138,147)
(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)(164,173)
(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(190,199)
(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)
( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)
( 51, 52)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 65, 66)( 67, 72)( 68, 71)
( 69, 70)( 74, 75)( 76, 81)( 77, 80)( 78, 79)( 83, 84)( 85, 90)( 86, 89)
( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)
(105,106)(109,136)(110,138)(111,137)(112,144)(113,143)(114,142)(115,141)
(116,140)(117,139)(118,145)(119,147)(120,146)(121,153)(122,152)(123,151)
(124,150)(125,149)(126,148)(127,154)(128,156)(129,155)(130,162)(131,161)
(132,160)(133,159)(134,158)(135,157)(163,190)(164,192)(165,191)(166,198)
(167,197)(168,196)(169,195)(170,194)(171,193)(172,199)(173,201)(174,200)
(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,208)(182,210)
(183,209)(184,216)(185,215)(186,214)(187,213)(188,212)(189,211);
s1 := Sym(216)!(  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,117)
(  8,116)(  9,115)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)
( 16,126)( 17,125)( 18,124)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)
( 24,128)( 25,135)( 26,134)( 27,133)( 28,139)( 29,141)( 30,140)( 31,136)
( 32,138)( 33,137)( 34,144)( 35,143)( 36,142)( 37,148)( 38,150)( 39,149)
( 40,145)( 41,147)( 42,146)( 43,153)( 44,152)( 45,151)( 46,157)( 47,159)
( 48,158)( 49,154)( 50,156)( 51,155)( 52,162)( 53,161)( 54,160)( 55,166)
( 56,168)( 57,167)( 58,163)( 59,165)( 60,164)( 61,171)( 62,170)( 63,169)
( 64,175)( 65,177)( 66,176)( 67,172)( 68,174)( 69,173)( 70,180)( 71,179)
( 72,178)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)( 79,189)
( 80,188)( 81,187)( 82,193)( 83,195)( 84,194)( 85,190)( 86,192)( 87,191)
( 88,198)( 89,197)( 90,196)( 91,202)( 92,204)( 93,203)( 94,199)( 95,201)
( 96,200)( 97,207)( 98,206)( 99,205)(100,211)(101,213)(102,212)(103,208)
(104,210)(105,209)(106,216)(107,215)(108,214);
s2 := Sym(216)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,163)(110,164)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)(119,182)
(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,172)
(128,173)(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)
(152,215)(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)
(160,205)(161,206)(162,207);
s3 := Sym(216)!(  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)
(  8, 17)(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)
( 34, 43)( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)
( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)
(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)
(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)
(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)
(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)
(198,207);
poly := sub<Sym(216)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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