Polytope of Type {12,34}

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This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,34}*816
Also Known As : {12,34|2}. if this polytope has another name.
Group : SmallGroup(816,132)
Rank : 3
Schlafli Type : {12,34}
Number of vertices, edges, etc : 12, 204, 34
Order of s0s1s2 : 204
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {12,34,2} of size 1632
Vertex Figure Of :
   {2,12,34} of size 1632
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,34}*408
   3-fold quotients : {4,34}*272
   6-fold quotients : {2,34}*136
   12-fold quotients : {2,17}*68
   17-fold quotients : {12,2}*48
   34-fold quotients : {6,2}*24
   51-fold quotients : {4,2}*16
   68-fold quotients : {3,2}*12
   102-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,34}*1632, {12,68}*1632
Irregular Quotients (of which this is a minimal cover):
   None.

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);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, 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(204)!( 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(204)!(  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(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)(103,104)(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);
poly := sub<Sym(204)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, 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 >;
References : None.
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