Polytope of Type {108,4,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {108,4,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,2289)
Rank : 4
Schlafli Type : {108,4,2}
Number of vertices, edges, etc : 108, 216, 4, 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 : {108,2,2}*864, {54,4,2}*864a
   3-fold quotients : {36,4,2}*576a
   4-fold quotients : {54,2,2}*432
   6-fold quotients : {36,2,2}*288, {18,4,2}*288a
   8-fold quotients : {27,2,2}*216
   9-fold quotients : {12,4,2}*192a
   12-fold quotients : {18,2,2}*144
   18-fold quotients : {12,2,2}*96, {6,4,2}*96a
   24-fold quotients : {9,2,2}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {6,2,2}*48
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope