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Polytope of Type {4,108,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,108,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,2289)
Rank : 4
Schlafli Type : {4,108,2}
Number of vertices, edges, etc : 4, 216, 108, 2
Order of s0s1s2s3 : 108
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 : {2,108,2}*864, {4,54,2}*864a
   3-fold quotients : {4,36,2}*576a
   4-fold quotients : {2,54,2}*432
   6-fold quotients : {2,36,2}*288, {4,18,2}*288a
   8-fold quotients : {2,27,2}*216
   9-fold quotients : {4,12,2}*192a
   12-fold quotients : {2,18,2}*144
   18-fold quotients : {2,12,2}*96, {4,6,2}*96a
   24-fold quotients : {2,9,2}*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 := (109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)
(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)
(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)
(133,160)(134,161)(135,162)(163,190)(164,191)(165,192)(166,193)(167,194)
(168,195)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)
(176,203)(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)(183,210)
(184,211)(185,212)(186,213)(187,214)(188,215)(189,216);;
s1 := (  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)(  8,113)
(  9,112)( 10,135)( 11,134)( 12,133)( 13,132)( 14,131)( 15,130)( 16,129)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,122)( 24,121)
( 25,120)( 26,119)( 27,118)( 28,136)( 29,138)( 30,137)( 31,144)( 32,143)
( 33,142)( 34,141)( 35,140)( 36,139)( 37,162)( 38,161)( 39,160)( 40,159)
( 41,158)( 42,157)( 43,156)( 44,155)( 45,154)( 46,153)( 47,152)( 48,151)
( 49,150)( 50,149)( 51,148)( 52,147)( 53,146)( 54,145)( 55,163)( 56,165)
( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)( 64,189)
( 65,188)( 66,187)( 67,186)( 68,185)( 69,184)( 70,183)( 71,182)( 72,181)
( 73,180)( 74,179)( 75,178)( 76,177)( 77,176)( 78,175)( 79,174)( 80,173)
( 81,172)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)( 88,195)
( 89,194)( 90,193)( 91,216)( 92,215)( 93,214)( 94,213)( 95,212)( 96,211)
( 97,210)( 98,209)( 99,208)(100,207)(101,206)(102,205)(103,204)(104,203)
(105,202)(106,201)(107,200)(108,199);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)
(  9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)( 30, 38)
( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)( 47, 53)
( 48, 52)( 49, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)( 60, 70)
( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82, 91)
( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)( 90, 94)
(100,108)(101,107)(102,106)(103,105)(109,172)(110,174)(111,173)(112,180)
(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)(119,165)(120,164)
(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,189)(128,188)
(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,199)
(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)
(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)(151,195)(152,194)
(153,193)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)
(161,209)(162,208);;
s3 := (217,218);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*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*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*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)!(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)
(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)
(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)
(132,159)(133,160)(134,161)(135,162)(163,190)(164,191)(165,192)(166,193)
(167,194)(168,195)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)
(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)
(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216);
s1 := Sym(218)!(  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)
(  8,113)(  9,112)( 10,135)( 11,134)( 12,133)( 13,132)( 14,131)( 15,130)
( 16,129)( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,122)
( 24,121)( 25,120)( 26,119)( 27,118)( 28,136)( 29,138)( 30,137)( 31,144)
( 32,143)( 33,142)( 34,141)( 35,140)( 36,139)( 37,162)( 38,161)( 39,160)
( 40,159)( 41,158)( 42,157)( 43,156)( 44,155)( 45,154)( 46,153)( 47,152)
( 48,151)( 49,150)( 50,149)( 51,148)( 52,147)( 53,146)( 54,145)( 55,163)
( 56,165)( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)
( 64,189)( 65,188)( 66,187)( 67,186)( 68,185)( 69,184)( 70,183)( 71,182)
( 72,181)( 73,180)( 74,179)( 75,178)( 76,177)( 77,176)( 78,175)( 79,174)
( 80,173)( 81,172)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)
( 88,195)( 89,194)( 90,193)( 91,216)( 92,215)( 93,214)( 94,213)( 95,212)
( 96,211)( 97,210)( 98,209)( 99,208)(100,207)(101,206)(102,205)(103,204)
(104,203)(105,202)(106,201)(107,200)(108,199);
s2 := Sym(218)!(  1, 10)(  2, 12)(  3, 11)(  4, 18)(  5, 17)(  6, 16)(  7, 15)
(  8, 14)(  9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)
( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)
( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)(100,108)(101,107)(102,106)(103,105)(109,172)(110,174)(111,173)
(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)(119,165)
(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,189)
(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)
(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)(143,203)
(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)(151,195)
(152,194)(153,193)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)
(160,210)(161,209)(162,208);
s3 := Sym(218)!(217,218);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*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*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*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 >; 
 

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