Polytope of Type {2,12,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,36}*1728a
if this polytope has a name.
Group : SmallGroup(1728,16615)
Rank : 4
Schlafli Type : {2,12,36}
Number of vertices, edges, etc : 2, 12, 216, 36
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 : {2,6,36}*864a, {2,12,18}*864a
   3-fold quotients : {2,4,36}*576a, {2,12,12}*576a
   4-fold quotients : {2,6,18}*432a
   6-fold quotients : {2,2,36}*288, {2,4,18}*288a, {2,6,12}*288a, {2,12,6}*288a
   9-fold quotients : {2,4,12}*192a, {2,12,4}*192a
   12-fold quotients : {2,2,18}*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 : {2,2,9}*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 := (  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 24, 27)( 25, 28)
( 26, 29)( 33, 36)( 34, 37)( 35, 38)( 42, 45)( 43, 46)( 44, 47)( 51, 54)
( 52, 55)( 53, 56)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)( 71, 74)
( 78, 81)( 79, 82)( 80, 83)( 87, 90)( 88, 91)( 89, 92)( 96, 99)( 97,100)
( 98,101)(105,108)(106,109)(107,110)(111,138)(112,139)(113,140)(114,144)
(115,145)(116,146)(117,141)(118,142)(119,143)(120,147)(121,148)(122,149)
(123,153)(124,154)(125,155)(126,150)(127,151)(128,152)(129,156)(130,157)
(131,158)(132,162)(133,163)(134,164)(135,159)(136,160)(137,161)(165,192)
(166,193)(167,194)(168,198)(169,199)(170,200)(171,195)(172,196)(173,197)
(174,201)(175,202)(176,203)(177,207)(178,208)(179,209)(180,204)(181,205)
(182,206)(183,210)(184,211)(185,212)(186,216)(187,217)(188,218)(189,213)
(190,214)(191,215);;
s2 := (  3,114)(  4,116)(  5,115)(  6,111)(  7,113)(  8,112)(  9,117)( 10,119)
( 11,118)( 12,134)( 13,133)( 14,132)( 15,131)( 16,130)( 17,129)( 18,137)
( 19,136)( 20,135)( 21,125)( 22,124)( 23,123)( 24,122)( 25,121)( 26,120)
( 27,128)( 28,127)( 29,126)( 30,141)( 31,143)( 32,142)( 33,138)( 34,140)
( 35,139)( 36,144)( 37,146)( 38,145)( 39,161)( 40,160)( 41,159)( 42,158)
( 43,157)( 44,156)( 45,164)( 46,163)( 47,162)( 48,152)( 49,151)( 50,150)
( 51,149)( 52,148)( 53,147)( 54,155)( 55,154)( 56,153)( 57,168)( 58,170)
( 59,169)( 60,165)( 61,167)( 62,166)( 63,171)( 64,173)( 65,172)( 66,188)
( 67,187)( 68,186)( 69,185)( 70,184)( 71,183)( 72,191)( 73,190)( 74,189)
( 75,179)( 76,178)( 77,177)( 78,176)( 79,175)( 80,174)( 81,182)( 82,181)
( 83,180)( 84,195)( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)( 90,198)
( 91,200)( 92,199)( 93,215)( 94,214)( 95,213)( 96,212)( 97,211)( 98,210)
( 99,218)(100,217)(101,216)(102,206)(103,205)(104,204)(105,203)(106,202)
(107,201)(108,209)(109,208)(110,207);;
s3 := (  3, 12)(  4, 14)(  5, 13)(  6, 15)(  7, 17)(  8, 16)(  9, 18)( 10, 20)
( 11, 19)( 21, 23)( 24, 26)( 27, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 42)
( 34, 44)( 35, 43)( 36, 45)( 37, 47)( 38, 46)( 48, 50)( 51, 53)( 54, 56)
( 57, 66)( 58, 68)( 59, 67)( 60, 69)( 61, 71)( 62, 70)( 63, 72)( 64, 74)
( 65, 73)( 75, 77)( 78, 80)( 81, 83)( 84, 93)( 85, 95)( 86, 94)( 87, 96)
( 88, 98)( 89, 97)( 90, 99)( 91,101)( 92,100)(102,104)(105,107)(108,110)
(111,174)(112,176)(113,175)(114,177)(115,179)(116,178)(117,180)(118,182)
(119,181)(120,165)(121,167)(122,166)(123,168)(124,170)(125,169)(126,171)
(127,173)(128,172)(129,185)(130,184)(131,183)(132,188)(133,187)(134,186)
(135,191)(136,190)(137,189)(138,201)(139,203)(140,202)(141,204)(142,206)
(143,205)(144,207)(145,209)(146,208)(147,192)(148,194)(149,193)(150,195)
(151,197)(152,196)(153,198)(154,200)(155,199)(156,212)(157,211)(158,210)
(159,215)(160,214)(161,213)(162,218)(163,217)(164,216);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope