Polytope of Type {2,36,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,36,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,16615)
Rank : 4
Schlafli Type : {2,36,12}
Number of vertices, edges, etc : 2, 36, 216, 12
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,36,6}*864a, {2,18,12}*864a
   3-fold quotients : {2,36,4}*576a, {2,12,12}*576a
   4-fold quotients : {2,18,6}*432a
   6-fold quotients : {2,36,2}*288, {2,18,4}*288a, {2,6,12}*288a, {2,12,6}*288a
   9-fold quotients : {2,4,12}*192a, {2,12,4}*192a
   12-fold quotients : {2,18,2}*144, {2,6,6}*144a
   18-fold quotients : {2,2,12}*96, {2,12,2}*96, {2,4,6}*96a, {2,6,4}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)( 16, 25)
( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)( 39, 50)
( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)( 47, 54)
( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)( 70, 79)
( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)( 93,104)
( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)(101,108)
(111,165)(112,167)(113,166)(114,168)(115,170)(116,169)(117,171)(118,173)
(119,172)(120,185)(121,184)(122,183)(123,188)(124,187)(125,186)(126,191)
(127,190)(128,189)(129,176)(130,175)(131,174)(132,179)(133,178)(134,177)
(135,182)(136,181)(137,180)(138,192)(139,194)(140,193)(141,195)(142,197)
(143,196)(144,198)(145,200)(146,199)(147,212)(148,211)(149,210)(150,215)
(151,214)(152,213)(153,218)(154,217)(155,216)(156,203)(157,202)(158,201)
(159,206)(160,205)(161,204)(162,209)(163,208)(164,207);;
s2 := (  3,120)(  4,122)(  5,121)(  6,126)(  7,128)(  8,127)(  9,123)( 10,125)
( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)( 18,114)
( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,153)( 34,155)
( 35,154)( 36,150)( 37,152)( 38,151)( 39,138)( 40,140)( 41,139)( 42,144)
( 43,146)( 44,145)( 45,141)( 46,143)( 47,142)( 48,158)( 49,157)( 50,156)
( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,174)( 58,176)
( 59,175)( 60,180)( 61,182)( 62,181)( 63,177)( 64,179)( 65,178)( 66,165)
( 67,167)( 68,166)( 69,171)( 70,173)( 71,172)( 72,168)( 73,170)( 74,169)
( 75,185)( 76,184)( 77,183)( 78,191)( 79,190)( 80,189)( 81,188)( 82,187)
( 83,186)( 84,201)( 85,203)( 86,202)( 87,207)( 88,209)( 89,208)( 90,204)
( 91,206)( 92,205)( 93,192)( 94,194)( 95,193)( 96,198)( 97,200)( 98,199)
( 99,195)(100,197)(101,196)(102,212)(103,211)(104,210)(105,218)(106,217)
(107,216)(108,215)(109,214)(110,213);;
s3 := (  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)( 22, 25)
( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)( 48, 51)
( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)( 68, 71)
( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)( 94, 97)
( 95, 98)(102,105)(103,106)(104,107)(111,141)(112,142)(113,143)(114,138)
(115,139)(116,140)(117,144)(118,145)(119,146)(120,150)(121,151)(122,152)
(123,147)(124,148)(125,149)(126,153)(127,154)(128,155)(129,159)(130,160)
(131,161)(132,156)(133,157)(134,158)(135,162)(136,163)(137,164)(165,195)
(166,196)(167,197)(168,192)(169,193)(170,194)(171,198)(172,199)(173,200)
(174,204)(175,205)(176,206)(177,201)(178,202)(179,203)(180,207)(181,208)
(182,209)(183,213)(184,214)(185,215)(186,210)(187,211)(188,212)(189,216)
(190,217)(191,218);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)
( 16, 25)( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)
( 39, 50)( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)
( 47, 54)( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)
( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 85, 86)( 88, 89)( 91, 92)
( 93,104)( 94,103)( 95,102)( 96,107)( 97,106)( 98,105)( 99,110)(100,109)
(101,108)(111,165)(112,167)(113,166)(114,168)(115,170)(116,169)(117,171)
(118,173)(119,172)(120,185)(121,184)(122,183)(123,188)(124,187)(125,186)
(126,191)(127,190)(128,189)(129,176)(130,175)(131,174)(132,179)(133,178)
(134,177)(135,182)(136,181)(137,180)(138,192)(139,194)(140,193)(141,195)
(142,197)(143,196)(144,198)(145,200)(146,199)(147,212)(148,211)(149,210)
(150,215)(151,214)(152,213)(153,218)(154,217)(155,216)(156,203)(157,202)
(158,201)(159,206)(160,205)(161,204)(162,209)(163,208)(164,207);
s2 := Sym(218)!(  3,120)(  4,122)(  5,121)(  6,126)(  7,128)(  8,127)(  9,123)
( 10,125)( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)
( 18,114)( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)
( 26,135)( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,153)
( 34,155)( 35,154)( 36,150)( 37,152)( 38,151)( 39,138)( 40,140)( 41,139)
( 42,144)( 43,146)( 44,145)( 45,141)( 46,143)( 47,142)( 48,158)( 49,157)
( 50,156)( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,174)
( 58,176)( 59,175)( 60,180)( 61,182)( 62,181)( 63,177)( 64,179)( 65,178)
( 66,165)( 67,167)( 68,166)( 69,171)( 70,173)( 71,172)( 72,168)( 73,170)
( 74,169)( 75,185)( 76,184)( 77,183)( 78,191)( 79,190)( 80,189)( 81,188)
( 82,187)( 83,186)( 84,201)( 85,203)( 86,202)( 87,207)( 88,209)( 89,208)
( 90,204)( 91,206)( 92,205)( 93,192)( 94,194)( 95,193)( 96,198)( 97,200)
( 98,199)( 99,195)(100,197)(101,196)(102,212)(103,211)(104,210)(105,218)
(106,217)(107,216)(108,215)(109,214)(110,213);
s3 := Sym(218)!(  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)
( 22, 25)( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)
( 48, 51)( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)
( 68, 71)( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)
( 94, 97)( 95, 98)(102,105)(103,106)(104,107)(111,141)(112,142)(113,143)
(114,138)(115,139)(116,140)(117,144)(118,145)(119,146)(120,150)(121,151)
(122,152)(123,147)(124,148)(125,149)(126,153)(127,154)(128,155)(129,159)
(130,160)(131,161)(132,156)(133,157)(134,158)(135,162)(136,163)(137,164)
(165,195)(166,196)(167,197)(168,192)(169,193)(170,194)(171,198)(172,199)
(173,200)(174,204)(175,205)(176,206)(177,201)(178,202)(179,203)(180,207)
(181,208)(182,209)(183,213)(184,214)(185,215)(186,210)(187,211)(188,212)
(189,216)(190,217)(191,218);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope