Polytope of Type {6,36,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,14461)
Rank : 4
Schlafli Type : {6,36,4}
Number of vertices, edges, etc : 6, 108, 72, 4
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 : {6,36,2}*864b, {6,18,4}*864b
   3-fold quotients : {2,36,4}*576a, {6,12,4}*576b
   4-fold quotients : {6,18,2}*432b
   6-fold quotients : {2,36,2}*288, {2,18,4}*288a, {6,12,2}*288b, {6,6,4}*288b
   8-fold quotients : {6,9,2}*216
   9-fold quotients : {2,12,4}*192a
   12-fold quotients : {2,18,2}*144, {6,6,2}*144b
   18-fold quotients : {2,12,2}*96, {2,6,4}*96a
   24-fold quotients : {2,9,2}*72, {6,3,2}*72
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
to this polytope