Polytope of Type {4,34,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,34,6}*1632
Also Known As : {{4,34|2},{34,6|2}}. if this polytope has another name.
Group : SmallGroup(1632,1097)
Rank : 4
Schlafli Type : {4,34,6}
Number of vertices, edges, etc : 4, 68, 102, 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 : {2,34,6}*816
   3-fold quotients : {4,34,2}*544
   6-fold quotients : {2,34,2}*272
   12-fold quotients : {2,17,2}*136
   17-fold quotients : {4,2,6}*96
   34-fold quotients : {4,2,3}*48, {2,2,6}*48
   51-fold quotients : {4,2,2}*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 := (103,154)(104,155)(105,156)(106,157)(107,158)(108,159)(109,160)(110,161)
(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)(118,169)
(119,170)(120,171)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)
(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)(134,185)
(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)(142,193)
(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)(150,201)
(151,202)(152,203)(153,204);;
s1 := (  1,103)(  2,119)(  3,118)(  4,117)(  5,116)(  6,115)(  7,114)(  8,113)
(  9,112)( 10,111)( 11,110)( 12,109)( 13,108)( 14,107)( 15,106)( 16,105)
( 17,104)( 18,120)( 19,136)( 20,135)( 21,134)( 22,133)( 23,132)( 24,131)
( 25,130)( 26,129)( 27,128)( 28,127)( 29,126)( 30,125)( 31,124)( 32,123)
( 33,122)( 34,121)( 35,137)( 36,153)( 37,152)( 38,151)( 39,150)( 40,149)
( 41,148)( 42,147)( 43,146)( 44,145)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,140)( 50,139)( 51,138)( 52,154)( 53,170)( 54,169)( 55,168)( 56,167)
( 57,166)( 58,165)( 59,164)( 60,163)( 61,162)( 62,161)( 63,160)( 64,159)
( 65,158)( 66,157)( 67,156)( 68,155)( 69,171)( 70,187)( 71,186)( 72,185)
( 73,184)( 74,183)( 75,182)( 76,181)( 77,180)( 78,179)( 79,178)( 80,177)
( 81,176)( 82,175)( 83,174)( 84,173)( 85,172)( 86,188)( 87,204)( 88,203)
( 89,202)( 90,201)( 91,200)( 92,199)( 93,198)( 94,197)( 95,196)( 96,195)
( 97,194)( 98,193)( 99,192)(100,191)(101,190)(102,189);;
s2 := (  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)
( 18, 36)( 19, 35)( 20, 51)( 21, 50)( 22, 49)( 23, 48)( 24, 47)( 25, 46)
( 26, 45)( 27, 44)( 28, 43)( 29, 42)( 30, 41)( 31, 40)( 32, 39)( 33, 38)
( 34, 37)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)
( 60, 62)( 69, 87)( 70, 86)( 71,102)( 72,101)( 73,100)( 74, 99)( 75, 98)
( 76, 97)( 77, 96)( 78, 95)( 79, 94)( 80, 93)( 81, 92)( 82, 91)( 83, 90)
( 84, 89)( 85, 88)(103,104)(105,119)(106,118)(107,117)(108,116)(109,115)
(110,114)(111,113)(120,138)(121,137)(122,153)(123,152)(124,151)(125,150)
(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)(133,142)
(134,141)(135,140)(136,139)(154,155)(156,170)(157,169)(158,168)(159,167)
(160,166)(161,165)(162,164)(171,189)(172,188)(173,204)(174,203)(175,202)
(176,201)(177,200)(178,199)(179,198)(180,197)(181,196)(182,195)(183,194)
(184,193)(185,192)(186,191)(187,190);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(204)!(103,154)(104,155)(105,156)(106,157)(107,158)(108,159)(109,160)
(110,161)(111,162)(112,163)(113,164)(114,165)(115,166)(116,167)(117,168)
(118,169)(119,170)(120,171)(121,172)(122,173)(123,174)(124,175)(125,176)
(126,177)(127,178)(128,179)(129,180)(130,181)(131,182)(132,183)(133,184)
(134,185)(135,186)(136,187)(137,188)(138,189)(139,190)(140,191)(141,192)
(142,193)(143,194)(144,195)(145,196)(146,197)(147,198)(148,199)(149,200)
(150,201)(151,202)(152,203)(153,204);
s1 := Sym(204)!(  1,103)(  2,119)(  3,118)(  4,117)(  5,116)(  6,115)(  7,114)
(  8,113)(  9,112)( 10,111)( 11,110)( 12,109)( 13,108)( 14,107)( 15,106)
( 16,105)( 17,104)( 18,120)( 19,136)( 20,135)( 21,134)( 22,133)( 23,132)
( 24,131)( 25,130)( 26,129)( 27,128)( 28,127)( 29,126)( 30,125)( 31,124)
( 32,123)( 33,122)( 34,121)( 35,137)( 36,153)( 37,152)( 38,151)( 39,150)
( 40,149)( 41,148)( 42,147)( 43,146)( 44,145)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,140)( 50,139)( 51,138)( 52,154)( 53,170)( 54,169)( 55,168)
( 56,167)( 57,166)( 58,165)( 59,164)( 60,163)( 61,162)( 62,161)( 63,160)
( 64,159)( 65,158)( 66,157)( 67,156)( 68,155)( 69,171)( 70,187)( 71,186)
( 72,185)( 73,184)( 74,183)( 75,182)( 76,181)( 77,180)( 78,179)( 79,178)
( 80,177)( 81,176)( 82,175)( 83,174)( 84,173)( 85,172)( 86,188)( 87,204)
( 88,203)( 89,202)( 90,201)( 91,200)( 92,199)( 93,198)( 94,197)( 95,196)
( 96,195)( 97,194)( 98,193)( 99,192)(100,191)(101,190)(102,189);
s2 := Sym(204)!(  1,  2)(  3, 17)(  4, 16)(  5, 15)(  6, 14)(  7, 13)(  8, 12)
(  9, 11)( 18, 36)( 19, 35)( 20, 51)( 21, 50)( 22, 49)( 23, 48)( 24, 47)
( 25, 46)( 26, 45)( 27, 44)( 28, 43)( 29, 42)( 30, 41)( 31, 40)( 32, 39)
( 33, 38)( 34, 37)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)
( 59, 63)( 60, 62)( 69, 87)( 70, 86)( 71,102)( 72,101)( 73,100)( 74, 99)
( 75, 98)( 76, 97)( 77, 96)( 78, 95)( 79, 94)( 80, 93)( 81, 92)( 82, 91)
( 83, 90)( 84, 89)( 85, 88)(103,104)(105,119)(106,118)(107,117)(108,116)
(109,115)(110,114)(111,113)(120,138)(121,137)(122,153)(123,152)(124,151)
(125,150)(126,149)(127,148)(128,147)(129,146)(130,145)(131,144)(132,143)
(133,142)(134,141)(135,140)(136,139)(154,155)(156,170)(157,169)(158,168)
(159,167)(160,166)(161,165)(162,164)(171,189)(172,188)(173,204)(174,203)
(175,202)(176,201)(177,200)(178,199)(179,198)(180,197)(181,196)(182,195)
(183,194)(184,193)(185,192)(186,191)(187,190);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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