Polytope of Type {12,34,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,34,2}*1632
if this polytope has a name.
Group : SmallGroup(1632,1087)
Rank : 4
Schlafli Type : {12,34,2}
Number of vertices, edges, etc : 12, 204, 34, 2
Order of s₀s₁s₂s₃ : 204
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 : {6,34,2}*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 : {12,2,2}*96
   34-fold quotients : {6,2,2}*48
   51-fold quotients : {4,2,2}*32
   68-fold quotients : {3,2,2}*24
   102-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 18, 35)( 19, 36)( 20, 37)( 21, 38)( 22, 39)( 23, 40)( 24, 41)( 25, 42)
( 26, 43)( 27, 44)( 28, 45)( 29, 46)( 30, 47)( 31, 48)( 32, 49)( 33, 50)
( 34, 51)( 69, 86)( 70, 87)( 71, 88)( 72, 89)( 73, 90)( 74, 91)( 75, 92)
( 76, 93)( 77, 94)( 78, 95)( 79, 96)( 80, 97)( 81, 98)( 82, 99)( 83,100)
( 84,101)( 85,102)(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)(120,188)(121,189)(122,190)(123,191)(124,192)
(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)(132,200)
(133,201)(134,202)(135,203)(136,204)(137,171)(138,172)(139,173)(140,174)
(141,175)(142,176)(143,177)(144,178)(145,179)(146,180)(147,181)(148,182)
(149,183)(150,184)(151,185)(152,186)(153,187);;
s1 := (  1,120)(  2,136)(  3,135)(  4,134)(  5,133)(  6,132)(  7,131)(  8,130)
(  9,129)( 10,128)( 11,127)( 12,126)( 13,125)( 14,124)( 15,123)( 16,122)
( 17,121)( 18,103)( 19,119)( 20,118)( 21,117)( 22,116)( 23,115)( 24,114)
( 25,113)( 26,112)( 27,111)( 28,110)( 29,109)( 30,108)( 31,107)( 32,106)
( 33,105)( 34,104)( 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)( 52,171)( 53,187)( 54,186)( 55,185)( 56,184)
( 57,183)( 58,182)( 59,181)( 60,180)( 61,179)( 62,178)( 63,177)( 64,176)
( 65,175)( 66,174)( 67,173)( 68,172)( 69,154)( 70,170)( 71,169)( 72,168)
( 73,167)( 74,166)( 75,165)( 76,164)( 77,163)( 78,162)( 79,161)( 80,160)
( 81,159)( 82,158)( 83,157)( 84,156)( 85,155)( 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, 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)
(103,104)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)
(120,121)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)
(137,138)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)
(154,155)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)
(171,172)(173,187)(174,186)(175,185)(176,184)(177,183)(178,182)(179,181)
(188,189)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198);;
s3 := (205,206);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(206)!( 18, 35)( 19, 36)( 20, 37)( 21, 38)( 22, 39)( 23, 40)( 24, 41)
( 25, 42)( 26, 43)( 27, 44)( 28, 45)( 29, 46)( 30, 47)( 31, 48)( 32, 49)
( 33, 50)( 34, 51)( 69, 86)( 70, 87)( 71, 88)( 72, 89)( 73, 90)( 74, 91)
( 75, 92)( 76, 93)( 77, 94)( 78, 95)( 79, 96)( 80, 97)( 81, 98)( 82, 99)
( 83,100)( 84,101)( 85,102)(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)(120,188)(121,189)(122,190)(123,191)
(124,192)(125,193)(126,194)(127,195)(128,196)(129,197)(130,198)(131,199)
(132,200)(133,201)(134,202)(135,203)(136,204)(137,171)(138,172)(139,173)
(140,174)(141,175)(142,176)(143,177)(144,178)(145,179)(146,180)(147,181)
(148,182)(149,183)(150,184)(151,185)(152,186)(153,187);
s1 := Sym(206)!(  1,120)(  2,136)(  3,135)(  4,134)(  5,133)(  6,132)(  7,131)
(  8,130)(  9,129)( 10,128)( 11,127)( 12,126)( 13,125)( 14,124)( 15,123)
( 16,122)( 17,121)( 18,103)( 19,119)( 20,118)( 21,117)( 22,116)( 23,115)
( 24,114)( 25,113)( 26,112)( 27,111)( 28,110)( 29,109)( 30,108)( 31,107)
( 32,106)( 33,105)( 34,104)( 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)( 52,171)( 53,187)( 54,186)( 55,185)
( 56,184)( 57,183)( 58,182)( 59,181)( 60,180)( 61,179)( 62,178)( 63,177)
( 64,176)( 65,175)( 66,174)( 67,173)( 68,172)( 69,154)( 70,170)( 71,169)
( 72,168)( 73,167)( 74,166)( 75,165)( 76,164)( 77,163)( 78,162)( 79,161)
( 80,160)( 81,159)( 82,158)( 83,157)( 84,156)( 85,155)( 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(206)!(  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)(103,104)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)
(111,113)(120,121)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)
(128,130)(137,138)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)
(145,147)(154,155)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)
(162,164)(171,172)(173,187)(174,186)(175,185)(176,184)(177,183)(178,182)
(179,181)(188,189)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)
(196,198);
s3 := Sym(206)!(205,206);
poly := sub<Sym(206)|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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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 >;

to this polytope