Polytope of Type {4,36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,14461)
Rank : 4
Schlafli Type : {4,36,6}
Number of vertices, edges, etc : 4, 72, 108, 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 : {2,36,6}*864b, {4,18,6}*864b
   3-fold quotients : {4,36,2}*576a, {4,12,6}*576b
   4-fold quotients : {2,18,6}*432b
   6-fold quotients : {2,36,2}*288, {4,18,2}*288a, {2,12,6}*288b, {4,6,6}*288b
   8-fold quotients : {2,9,6}*216
   9-fold quotients : {4,12,2}*192a
   12-fold quotients : {2,18,2}*144, {2,6,6}*144c
   18-fold quotients : {2,12,2}*96, {4,6,2}*96a
   24-fold quotients : {2,9,2}*72, {2,3,6}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,4,2}*32, {4,2,2}*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 := (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);;
s1 := (  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)(  8,113)
(  9,112)( 10,127)( 11,129)( 12,128)( 13,135)( 14,134)( 15,133)( 16,132)
( 17,131)( 18,130)( 19,118)( 20,120)( 21,119)( 22,126)( 23,125)( 24,124)
( 25,123)( 26,122)( 27,121)( 28,136)( 29,138)( 30,137)( 31,144)( 32,143)
( 33,142)( 34,141)( 35,140)( 36,139)( 37,154)( 38,156)( 39,155)( 40,162)
( 41,161)( 42,160)( 43,159)( 44,158)( 45,157)( 46,145)( 47,147)( 48,146)
( 49,153)( 50,152)( 51,151)( 52,150)( 53,149)( 54,148)( 55,163)( 56,165)
( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)( 64,181)
( 65,183)( 66,182)( 67,189)( 68,188)( 69,187)( 70,186)( 71,185)( 72,184)
( 73,172)( 74,174)( 75,173)( 76,180)( 77,179)( 78,178)( 79,177)( 80,176)
( 81,175)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)( 88,195)
( 89,194)( 90,193)( 91,208)( 92,210)( 93,209)( 94,216)( 95,215)( 96,214)
( 97,213)( 98,212)( 99,211)(100,199)(101,201)(102,200)(103,207)(104,206)
(105,205)(106,204)(107,203)(108,202);;
s2 := (  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 18)(  8, 17)
(  9, 16)( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 40)( 29, 42)( 30, 41)
( 31, 37)( 32, 39)( 33, 38)( 34, 45)( 35, 44)( 36, 43)( 46, 49)( 47, 51)
( 48, 50)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)
( 61, 72)( 62, 71)( 63, 70)( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 94)
( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 99)( 89, 98)( 90, 97)
(100,103)(101,105)(102,104)(106,108)(109,148)(110,150)(111,149)(112,145)
(113,147)(114,146)(115,153)(116,152)(117,151)(118,139)(119,141)(120,140)
(121,136)(122,138)(123,137)(124,144)(125,143)(126,142)(127,157)(128,159)
(129,158)(130,154)(131,156)(132,155)(133,162)(134,161)(135,160)(163,202)
(164,204)(165,203)(166,199)(167,201)(168,200)(169,207)(170,206)(171,205)
(172,193)(173,195)(174,194)(175,190)(176,192)(177,191)(178,198)(179,197)
(180,196)(181,211)(182,213)(183,212)(184,208)(185,210)(186,209)(187,216)
(188,215)(189,214);;
s3 := ( 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);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*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*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)!(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);
s1 := Sym(216)!(  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)
(  8,113)(  9,112)( 10,127)( 11,129)( 12,128)( 13,135)( 14,134)( 15,133)
( 16,132)( 17,131)( 18,130)( 19,118)( 20,120)( 21,119)( 22,126)( 23,125)
( 24,124)( 25,123)( 26,122)( 27,121)( 28,136)( 29,138)( 30,137)( 31,144)
( 32,143)( 33,142)( 34,141)( 35,140)( 36,139)( 37,154)( 38,156)( 39,155)
( 40,162)( 41,161)( 42,160)( 43,159)( 44,158)( 45,157)( 46,145)( 47,147)
( 48,146)( 49,153)( 50,152)( 51,151)( 52,150)( 53,149)( 54,148)( 55,163)
( 56,165)( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)
( 64,181)( 65,183)( 66,182)( 67,189)( 68,188)( 69,187)( 70,186)( 71,185)
( 72,184)( 73,172)( 74,174)( 75,173)( 76,180)( 77,179)( 78,178)( 79,177)
( 80,176)( 81,175)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)
( 88,195)( 89,194)( 90,193)( 91,208)( 92,210)( 93,209)( 94,216)( 95,215)
( 96,214)( 97,213)( 98,212)( 99,211)(100,199)(101,201)(102,200)(103,207)
(104,206)(105,205)(106,204)(107,203)(108,202);
s2 := Sym(216)!(  1, 13)(  2, 15)(  3, 14)(  4, 10)(  5, 12)(  6, 11)(  7, 18)
(  8, 17)(  9, 16)( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 40)( 29, 42)
( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 45)( 35, 44)( 36, 43)( 46, 49)
( 47, 51)( 48, 50)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)
( 60, 65)( 61, 72)( 62, 71)( 63, 70)( 73, 76)( 74, 78)( 75, 77)( 79, 81)
( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 99)( 89, 98)
( 90, 97)(100,103)(101,105)(102,104)(106,108)(109,148)(110,150)(111,149)
(112,145)(113,147)(114,146)(115,153)(116,152)(117,151)(118,139)(119,141)
(120,140)(121,136)(122,138)(123,137)(124,144)(125,143)(126,142)(127,157)
(128,159)(129,158)(130,154)(131,156)(132,155)(133,162)(134,161)(135,160)
(163,202)(164,204)(165,203)(166,199)(167,201)(168,200)(169,207)(170,206)
(171,205)(172,193)(173,195)(174,194)(175,190)(176,192)(177,191)(178,198)
(179,197)(180,196)(181,211)(182,213)(183,212)(184,208)(185,210)(186,209)
(187,216)(188,215)(189,214);
s3 := 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);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*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*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