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Polytope of Type {34,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {34,4,6}*1632
Also Known As : {{34,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(1632,1097)
Rank : 4
Schlafli Type : {34,4,6}
Number of vertices, edges, etc : 34, 68, 12, 6
Order of s0s1s2s3 : 204
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 : {34,2,6}*816
   3-fold quotients : {34,4,2}*544
   4-fold quotients : {17,2,6}*408, {34,2,3}*408
   6-fold quotients : {34,2,2}*272
   8-fold quotients : {17,2,3}*204
   12-fold quotients : {17,2,2}*136
   17-fold quotients : {2,4,6}*96a
   34-fold quotients : {2,2,6}*48
   51-fold quotients : {2,4,2}*32
   68-fold quotients : {2,2,3}*24
   102-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)
( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 25, 28)( 26, 27)
( 36, 51)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)( 43, 44)
( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)( 60, 61)
( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)( 77, 78)
( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)
(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)
(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)
(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)(145,146)
(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)
(172,187)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)(179,180)
(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197);;
s1 := (  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)
( 18, 19)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)( 26, 28)
( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 45)
( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)( 60, 62)
( 69, 70)( 71, 85)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)
( 86, 87)( 88,102)( 89,101)( 90,100)( 91, 99)( 92, 98)( 93, 97)( 94, 96)
(103,155)(104,154)(105,170)(106,169)(107,168)(108,167)(109,166)(110,165)
(111,164)(112,163)(113,162)(114,161)(115,160)(116,159)(117,158)(118,157)
(119,156)(120,172)(121,171)(122,187)(123,186)(124,185)(125,184)(126,183)
(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)(134,175)
(135,174)(136,173)(137,189)(138,188)(139,204)(140,203)(141,202)(142,201)
(143,200)(144,199)(145,198)(146,197)(147,196)(148,195)(149,194)(150,193)
(151,192)(152,191)(153,190);;
s2 := (  1,103)(  2,104)(  3,105)(  4,106)(  5,107)(  6,108)(  7,109)(  8,110)
(  9,111)( 10,112)( 11,113)( 12,114)( 13,115)( 14,116)( 15,117)( 16,118)
( 17,119)( 18,137)( 19,138)( 20,139)( 21,140)( 22,141)( 23,142)( 24,143)
( 25,144)( 26,145)( 27,146)( 28,147)( 29,148)( 30,149)( 31,150)( 32,151)
( 33,152)( 34,153)( 35,120)( 36,121)( 37,122)( 38,123)( 39,124)( 40,125)
( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,131)( 47,132)( 48,133)
( 49,134)( 50,135)( 51,136)( 52,154)( 53,155)( 54,156)( 55,157)( 56,158)
( 57,159)( 58,160)( 59,161)( 60,162)( 61,163)( 62,164)( 63,165)( 64,166)
( 65,167)( 66,168)( 67,169)( 68,170)( 69,188)( 70,189)( 71,190)( 72,191)
( 73,192)( 74,193)( 75,194)( 76,195)( 77,196)( 78,197)( 79,198)( 80,199)
( 81,200)( 82,201)( 83,202)( 84,203)( 85,204)( 86,171)( 87,172)( 88,173)
( 89,174)( 90,175)( 91,176)( 92,177)( 93,178)( 94,179)( 95,180)( 96,181)
( 97,182)( 98,183)( 99,184)(100,185)(101,186)(102,187);;
s3 := (  1, 18)(  2, 19)(  3, 20)(  4, 21)(  5, 22)(  6, 23)(  7, 24)(  8, 25)
(  9, 26)( 10, 27)( 11, 28)( 12, 29)( 13, 30)( 14, 31)( 15, 32)( 16, 33)
( 17, 34)( 52, 69)( 53, 70)( 54, 71)( 55, 72)( 56, 73)( 57, 74)( 58, 75)
( 59, 76)( 60, 77)( 61, 78)( 62, 79)( 63, 80)( 64, 81)( 65, 82)( 66, 83)
( 67, 84)( 68, 85)(103,120)(104,121)(105,122)(106,123)(107,124)(108,125)
(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)(116,133)
(117,134)(118,135)(119,136)(154,171)(155,172)(156,173)(157,174)(158,175)
(159,176)(160,177)(161,178)(162,179)(163,180)(164,181)(165,182)(166,183)
(167,184)(168,185)(169,186)(170,187);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(204)!(  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)
(  9, 10)( 19, 34)( 20, 33)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 25, 28)
( 26, 27)( 36, 51)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 43, 44)( 53, 68)( 54, 67)( 55, 66)( 56, 65)( 57, 64)( 58, 63)( 59, 62)
( 60, 61)( 70, 85)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 80)( 76, 79)
( 77, 78)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)( 93, 96)
( 94, 95)(104,119)(105,118)(106,117)(107,116)(108,115)(109,114)(110,113)
(111,112)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)
(128,129)(138,153)(139,152)(140,151)(141,150)(142,149)(143,148)(144,147)
(145,146)(155,170)(156,169)(157,168)(158,167)(159,166)(160,165)(161,164)
(162,163)(172,187)(173,186)(174,185)(175,184)(176,183)(177,182)(178,181)
(179,180)(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)
(196,197);
s1 := Sym(204)!(  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)
(  9, 11)( 18, 19)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)
( 26, 28)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)
( 43, 45)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)
( 60, 62)( 69, 70)( 71, 85)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 86, 87)( 88,102)( 89,101)( 90,100)( 91, 99)( 92, 98)( 93, 97)
( 94, 96)(103,155)(104,154)(105,170)(106,169)(107,168)(108,167)(109,166)
(110,165)(111,164)(112,163)(113,162)(114,161)(115,160)(116,159)(117,158)
(118,157)(119,156)(120,172)(121,171)(122,187)(123,186)(124,185)(125,184)
(126,183)(127,182)(128,181)(129,180)(130,179)(131,178)(132,177)(133,176)
(134,175)(135,174)(136,173)(137,189)(138,188)(139,204)(140,203)(141,202)
(142,201)(143,200)(144,199)(145,198)(146,197)(147,196)(148,195)(149,194)
(150,193)(151,192)(152,191)(153,190);
s2 := Sym(204)!(  1,103)(  2,104)(  3,105)(  4,106)(  5,107)(  6,108)(  7,109)
(  8,110)(  9,111)( 10,112)( 11,113)( 12,114)( 13,115)( 14,116)( 15,117)
( 16,118)( 17,119)( 18,137)( 19,138)( 20,139)( 21,140)( 22,141)( 23,142)
( 24,143)( 25,144)( 26,145)( 27,146)( 28,147)( 29,148)( 30,149)( 31,150)
( 32,151)( 33,152)( 34,153)( 35,120)( 36,121)( 37,122)( 38,123)( 39,124)
( 40,125)( 41,126)( 42,127)( 43,128)( 44,129)( 45,130)( 46,131)( 47,132)
( 48,133)( 49,134)( 50,135)( 51,136)( 52,154)( 53,155)( 54,156)( 55,157)
( 56,158)( 57,159)( 58,160)( 59,161)( 60,162)( 61,163)( 62,164)( 63,165)
( 64,166)( 65,167)( 66,168)( 67,169)( 68,170)( 69,188)( 70,189)( 71,190)
( 72,191)( 73,192)( 74,193)( 75,194)( 76,195)( 77,196)( 78,197)( 79,198)
( 80,199)( 81,200)( 82,201)( 83,202)( 84,203)( 85,204)( 86,171)( 87,172)
( 88,173)( 89,174)( 90,175)( 91,176)( 92,177)( 93,178)( 94,179)( 95,180)
( 96,181)( 97,182)( 98,183)( 99,184)(100,185)(101,186)(102,187);
s3 := Sym(204)!(  1, 18)(  2, 19)(  3, 20)(  4, 21)(  5, 22)(  6, 23)(  7, 24)
(  8, 25)(  9, 26)( 10, 27)( 11, 28)( 12, 29)( 13, 30)( 14, 31)( 15, 32)
( 16, 33)( 17, 34)( 52, 69)( 53, 70)( 54, 71)( 55, 72)( 56, 73)( 57, 74)
( 58, 75)( 59, 76)( 60, 77)( 61, 78)( 62, 79)( 63, 80)( 64, 81)( 65, 82)
( 66, 83)( 67, 84)( 68, 85)(103,120)(104,121)(105,122)(106,123)(107,124)
(108,125)(109,126)(110,127)(111,128)(112,129)(113,130)(114,131)(115,132)
(116,133)(117,134)(118,135)(119,136)(154,171)(155,172)(156,173)(157,174)
(158,175)(159,176)(160,177)(161,178)(162,179)(163,180)(164,181)(165,182)
(166,183)(167,184)(168,185)(169,186)(170,187);
poly := sub<Sym(204)|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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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 >; 
 
References : None.
to this polytope