Polytope of Type {34,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {34,12}*816
Also Known As : {34,12|2}. if this polytope has another name.
Group : SmallGroup(816,132)
Rank : 3
Schlafli Type : {34,12}
Number of vertices, edges, etc : 34, 204, 12
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 :
   {34,12,2} of size 1632
Vertex Figure Of :
   {2,34,12} of size 1632
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {34,6}*408
   3-fold quotients : {34,4}*272
   6-fold quotients : {34,2}*136
   12-fold quotients : {17,2}*68
   17-fold quotients : {2,12}*48
   34-fold quotients : {2,6}*24
   51-fold quotients : {2,4}*16
   68-fold quotients : {2,3}*12
   102-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {34,24}*1632, {68,12}*1632
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, 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)(103,155)(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)(120,189)(121,188)(122,204)(123,203)(124,202)
(125,201)(126,200)(127,199)(128,198)(129,197)(130,196)(131,195)(132,194)
(133,193)(134,192)(135,191)(136,190)(137,172)(138,171)(139,187)(140,186)
(141,185)(142,184)(143,183)(144,182)(145,181)(146,180)(147,179)(148,178)
(149,177)(150,176)(151,175)(152,174)(153,173);;
s2 := (  1,120)(  2,121)(  3,122)(  4,123)(  5,124)(  6,125)(  7,126)(  8,127)
(  9,128)( 10,129)( 11,130)( 12,131)( 13,132)( 14,133)( 15,134)( 16,135)
( 17,136)( 18,103)( 19,104)( 20,105)( 21,106)( 22,107)( 23,108)( 24,109)
( 25,110)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,116)( 32,117)
( 33,118)( 34,119)( 35,137)( 36,138)( 37,139)( 38,140)( 39,141)( 40,142)
( 41,143)( 42,144)( 43,145)( 44,146)( 45,147)( 46,148)( 47,149)( 48,150)
( 49,151)( 50,152)( 51,153)( 52,171)( 53,172)( 54,173)( 55,174)( 56,175)
( 57,176)( 58,177)( 59,178)( 60,179)( 61,180)( 62,181)( 63,182)( 64,183)
( 65,184)( 66,185)( 67,186)( 68,187)( 69,154)( 70,155)( 71,156)( 72,157)
( 73,158)( 74,159)( 75,160)( 76,161)( 77,162)( 78,163)( 79,164)( 80,165)
( 81,166)( 82,167)( 83,168)( 84,169)( 85,170)( 86,188)( 87,189)( 88,190)
( 89,191)( 90,192)( 91,193)( 92,194)( 93,195)( 94,196)( 95,197)( 96,198)
( 97,199)( 98,200)( 99,201)(100,202)(101,203)(102,204);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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, 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)(103,155)(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)(120,189)(121,188)(122,204)(123,203)
(124,202)(125,201)(126,200)(127,199)(128,198)(129,197)(130,196)(131,195)
(132,194)(133,193)(134,192)(135,191)(136,190)(137,172)(138,171)(139,187)
(140,186)(141,185)(142,184)(143,183)(144,182)(145,181)(146,180)(147,179)
(148,178)(149,177)(150,176)(151,175)(152,174)(153,173);
s2 := Sym(204)!(  1,120)(  2,121)(  3,122)(  4,123)(  5,124)(  6,125)(  7,126)
(  8,127)(  9,128)( 10,129)( 11,130)( 12,131)( 13,132)( 14,133)( 15,134)
( 16,135)( 17,136)( 18,103)( 19,104)( 20,105)( 21,106)( 22,107)( 23,108)
( 24,109)( 25,110)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,116)
( 32,117)( 33,118)( 34,119)( 35,137)( 36,138)( 37,139)( 38,140)( 39,141)
( 40,142)( 41,143)( 42,144)( 43,145)( 44,146)( 45,147)( 46,148)( 47,149)
( 48,150)( 49,151)( 50,152)( 51,153)( 52,171)( 53,172)( 54,173)( 55,174)
( 56,175)( 57,176)( 58,177)( 59,178)( 60,179)( 61,180)( 62,181)( 63,182)
( 64,183)( 65,184)( 66,185)( 67,186)( 68,187)( 69,154)( 70,155)( 71,156)
( 72,157)( 73,158)( 74,159)( 75,160)( 76,161)( 77,162)( 78,163)( 79,164)
( 80,165)( 81,166)( 82,167)( 83,168)( 84,169)( 85,170)( 86,188)( 87,189)
( 88,190)( 89,191)( 90,192)( 91,193)( 92,194)( 93,195)( 94,196)( 95,197)
( 96,198)( 97,199)( 98,200)( 99,201)(100,202)(101,203)(102,204);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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