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Polytope of Type {2,2,27,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,27,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,20782)
Rank : 5
Schlafli Type : {2,2,27,4}
Number of vertices, edges, etc : 2, 2, 54, 108, 8
Order of s0s1s2s3s4 : 54
Order of s0s1s2s3s4s3s2s1 : 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 : {2,2,27,4}*864
   3-fold quotients : {2,2,9,4}*576
   4-fold quotients : {2,2,27,2}*432
   6-fold quotients : {2,2,9,4}*288
   9-fold quotients : {2,2,3,4}*192
   12-fold quotients : {2,2,9,2}*144
   18-fold quotients : {2,2,3,4}*96
   36-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := (  6,  7)(  9, 13)( 10, 15)( 11, 14)( 12, 16)( 17, 33)( 18, 35)( 19, 34)
( 20, 36)( 21, 29)( 22, 31)( 23, 30)( 24, 32)( 25, 37)( 26, 39)( 27, 38)
( 28, 40)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 97)( 46, 99)( 47, 98)
( 48,100)( 49, 93)( 50, 95)( 51, 94)( 52, 96)( 53, 77)( 54, 79)( 55, 78)
( 56, 80)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 81)( 62, 83)( 63, 82)
( 64, 84)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)
( 72,104)( 73,109)( 74,111)( 75,110)( 76,112)(114,115)(117,121)(118,123)
(119,122)(120,124)(125,141)(126,143)(127,142)(128,144)(129,137)(130,139)
(131,138)(132,140)(133,145)(134,147)(135,146)(136,148)(149,197)(150,199)
(151,198)(152,200)(153,205)(154,207)(155,206)(156,208)(157,201)(158,203)
(159,202)(160,204)(161,185)(162,187)(163,186)(164,188)(165,193)(166,195)
(167,194)(168,196)(169,189)(170,191)(171,190)(172,192)(173,213)(174,215)
(175,214)(176,216)(177,209)(178,211)(179,210)(180,212)(181,217)(182,219)
(183,218)(184,220);;
s3 := (  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 49)( 10, 50)( 11, 52)( 12, 51)
( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 69)( 18, 70)( 19, 72)( 20, 71)
( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 73)( 26, 74)( 27, 76)( 28, 75)
( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)( 36, 55)
( 37, 61)( 38, 62)( 39, 64)( 40, 63)( 77, 89)( 78, 90)( 79, 92)( 80, 91)
( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)( 88, 95)
(101,105)(102,106)(103,108)(104,107)(111,112)(113,149)(114,150)(115,152)
(116,151)(117,157)(118,158)(119,160)(120,159)(121,153)(122,154)(123,156)
(124,155)(125,177)(126,178)(127,180)(128,179)(129,173)(130,174)(131,176)
(132,175)(133,181)(134,182)(135,184)(136,183)(137,165)(138,166)(139,168)
(140,167)(141,161)(142,162)(143,164)(144,163)(145,169)(146,170)(147,172)
(148,171)(185,197)(186,198)(187,200)(188,199)(189,205)(190,206)(191,208)
(192,207)(193,201)(194,202)(195,204)(196,203)(209,213)(210,214)(211,216)
(212,215)(219,220);;
s4 := (  5,116)(  6,115)(  7,114)(  8,113)(  9,120)( 10,119)( 11,118)( 12,117)
( 13,124)( 14,123)( 15,122)( 16,121)( 17,128)( 18,127)( 19,126)( 20,125)
( 21,132)( 22,131)( 23,130)( 24,129)( 25,136)( 26,135)( 27,134)( 28,133)
( 29,140)( 30,139)( 31,138)( 32,137)( 33,144)( 34,143)( 35,142)( 36,141)
( 37,148)( 38,147)( 39,146)( 40,145)( 41,152)( 42,151)( 43,150)( 44,149)
( 45,156)( 46,155)( 47,154)( 48,153)( 49,160)( 50,159)( 51,158)( 52,157)
( 53,164)( 54,163)( 55,162)( 56,161)( 57,168)( 58,167)( 59,166)( 60,165)
( 61,172)( 62,171)( 63,170)( 64,169)( 65,176)( 66,175)( 67,174)( 68,173)
( 69,180)( 70,179)( 71,178)( 72,177)( 73,184)( 74,183)( 75,182)( 76,181)
( 77,188)( 78,187)( 79,186)( 80,185)( 81,192)( 82,191)( 83,190)( 84,189)
( 85,196)( 86,195)( 87,194)( 88,193)( 89,200)( 90,199)( 91,198)( 92,197)
( 93,204)( 94,203)( 95,202)( 96,201)( 97,208)( 98,207)( 99,206)(100,205)
(101,212)(102,211)(103,210)(104,209)(105,216)(106,215)(107,214)(108,213)
(109,220)(110,219)(111,218)(112,217);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*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(220)!(1,2);
s1 := Sym(220)!(3,4);
s2 := Sym(220)!(  6,  7)(  9, 13)( 10, 15)( 11, 14)( 12, 16)( 17, 33)( 18, 35)
( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)( 24, 32)( 25, 37)( 26, 39)
( 27, 38)( 28, 40)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 97)( 46, 99)
( 47, 98)( 48,100)( 49, 93)( 50, 95)( 51, 94)( 52, 96)( 53, 77)( 54, 79)
( 55, 78)( 56, 80)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 81)( 62, 83)
( 63, 82)( 64, 84)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)
( 71,102)( 72,104)( 73,109)( 74,111)( 75,110)( 76,112)(114,115)(117,121)
(118,123)(119,122)(120,124)(125,141)(126,143)(127,142)(128,144)(129,137)
(130,139)(131,138)(132,140)(133,145)(134,147)(135,146)(136,148)(149,197)
(150,199)(151,198)(152,200)(153,205)(154,207)(155,206)(156,208)(157,201)
(158,203)(159,202)(160,204)(161,185)(162,187)(163,186)(164,188)(165,193)
(166,195)(167,194)(168,196)(169,189)(170,191)(171,190)(172,192)(173,213)
(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212)(181,217)
(182,219)(183,218)(184,220);
s3 := Sym(220)!(  5, 41)(  6, 42)(  7, 44)(  8, 43)(  9, 49)( 10, 50)( 11, 52)
( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 69)( 18, 70)( 19, 72)
( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)( 25, 73)( 26, 74)( 27, 76)
( 28, 75)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)
( 36, 55)( 37, 61)( 38, 62)( 39, 64)( 40, 63)( 77, 89)( 78, 90)( 79, 92)
( 80, 91)( 81, 97)( 82, 98)( 83,100)( 84, 99)( 85, 93)( 86, 94)( 87, 96)
( 88, 95)(101,105)(102,106)(103,108)(104,107)(111,112)(113,149)(114,150)
(115,152)(116,151)(117,157)(118,158)(119,160)(120,159)(121,153)(122,154)
(123,156)(124,155)(125,177)(126,178)(127,180)(128,179)(129,173)(130,174)
(131,176)(132,175)(133,181)(134,182)(135,184)(136,183)(137,165)(138,166)
(139,168)(140,167)(141,161)(142,162)(143,164)(144,163)(145,169)(146,170)
(147,172)(148,171)(185,197)(186,198)(187,200)(188,199)(189,205)(190,206)
(191,208)(192,207)(193,201)(194,202)(195,204)(196,203)(209,213)(210,214)
(211,216)(212,215)(219,220);
s4 := Sym(220)!(  5,116)(  6,115)(  7,114)(  8,113)(  9,120)( 10,119)( 11,118)
( 12,117)( 13,124)( 14,123)( 15,122)( 16,121)( 17,128)( 18,127)( 19,126)
( 20,125)( 21,132)( 22,131)( 23,130)( 24,129)( 25,136)( 26,135)( 27,134)
( 28,133)( 29,140)( 30,139)( 31,138)( 32,137)( 33,144)( 34,143)( 35,142)
( 36,141)( 37,148)( 38,147)( 39,146)( 40,145)( 41,152)( 42,151)( 43,150)
( 44,149)( 45,156)( 46,155)( 47,154)( 48,153)( 49,160)( 50,159)( 51,158)
( 52,157)( 53,164)( 54,163)( 55,162)( 56,161)( 57,168)( 58,167)( 59,166)
( 60,165)( 61,172)( 62,171)( 63,170)( 64,169)( 65,176)( 66,175)( 67,174)
( 68,173)( 69,180)( 70,179)( 71,178)( 72,177)( 73,184)( 74,183)( 75,182)
( 76,181)( 77,188)( 78,187)( 79,186)( 80,185)( 81,192)( 82,191)( 83,190)
( 84,189)( 85,196)( 86,195)( 87,194)( 88,193)( 89,200)( 90,199)( 91,198)
( 92,197)( 93,204)( 94,203)( 95,202)( 96,201)( 97,208)( 98,207)( 99,206)
(100,205)(101,212)(102,211)(103,210)(104,209)(105,216)(106,215)(107,214)
(108,213)(109,220)(110,219)(111,218)(112,217);
poly := sub<Sym(220)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4*s3*s4, s2*s3*s4*s3*s2*s3*s2*s3*s4*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 >; 
 

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