Polytope of Type {2,68,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,68,6}*1632a
if this polytope has a name.
Group : SmallGroup(1632,1088)
Rank : 4
Schlafli Type : {2,68,6}
Number of vertices, edges, etc : 2, 68, 204, 6
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 : {2,34,6}*816
   3-fold quotients : {2,68,2}*544
   6-fold quotients : {2,34,2}*272
   12-fold quotients : {2,17,2}*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 := (1,2);;
s1 := (  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)( 11, 12)
( 21, 36)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)
( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 45, 46)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 72, 87)( 73, 86)( 74, 85)( 75, 84)( 76, 83)( 77, 82)( 78, 81)( 79, 80)
( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)( 95, 98)( 96, 97)
(105,156)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)(112,166)
(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)(120,158)
(121,157)(122,173)(123,189)(124,188)(125,187)(126,186)(127,185)(128,184)
(129,183)(130,182)(131,181)(132,180)(133,179)(134,178)(135,177)(136,176)
(137,175)(138,174)(139,190)(140,206)(141,205)(142,204)(143,203)(144,202)
(145,201)(146,200)(147,199)(148,198)(149,197)(150,196)(151,195)(152,194)
(153,193)(154,192)(155,191);;
s2 := (  3,106)(  4,105)(  5,121)(  6,120)(  7,119)(  8,118)(  9,117)( 10,116)
( 11,115)( 12,114)( 13,113)( 14,112)( 15,111)( 16,110)( 17,109)( 18,108)
( 19,107)( 20,140)( 21,139)( 22,155)( 23,154)( 24,153)( 25,152)( 26,151)
( 27,150)( 28,149)( 29,148)( 30,147)( 31,146)( 32,145)( 33,144)( 34,143)
( 35,142)( 36,141)( 37,123)( 38,122)( 39,138)( 40,137)( 41,136)( 42,135)
( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)( 48,129)( 49,128)( 50,127)
( 51,126)( 52,125)( 53,124)( 54,157)( 55,156)( 56,172)( 57,171)( 58,170)
( 59,169)( 60,168)( 61,167)( 62,166)( 63,165)( 64,164)( 65,163)( 66,162)
( 67,161)( 68,160)( 69,159)( 70,158)( 71,191)( 72,190)( 73,206)( 74,205)
( 75,204)( 76,203)( 77,202)( 78,201)( 79,200)( 80,199)( 81,198)( 82,197)
( 83,196)( 84,195)( 85,194)( 86,193)( 87,192)( 88,174)( 89,173)( 90,189)
( 91,188)( 92,187)( 93,186)( 94,185)( 95,184)( 96,183)( 97,182)( 98,181)
( 99,180)(100,179)(101,178)(102,177)(103,176)(104,175);;
s3 := (  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)( 18, 35)
( 19, 36)( 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)
( 69, 86)( 70, 87)(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)(120,137)(121,138)(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)(171,188)(172,189);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
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*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)!(1,2);
s1 := Sym(206)!(  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)
( 11, 12)( 21, 36)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 31)( 27, 30)
( 28, 29)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)
( 45, 46)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 72, 87)( 73, 86)( 74, 85)( 75, 84)( 76, 83)( 77, 82)( 78, 81)
( 79, 80)( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)( 95, 98)
( 96, 97)(105,156)(106,172)(107,171)(108,170)(109,169)(110,168)(111,167)
(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)(119,159)
(120,158)(121,157)(122,173)(123,189)(124,188)(125,187)(126,186)(127,185)
(128,184)(129,183)(130,182)(131,181)(132,180)(133,179)(134,178)(135,177)
(136,176)(137,175)(138,174)(139,190)(140,206)(141,205)(142,204)(143,203)
(144,202)(145,201)(146,200)(147,199)(148,198)(149,197)(150,196)(151,195)
(152,194)(153,193)(154,192)(155,191);
s2 := Sym(206)!(  3,106)(  4,105)(  5,121)(  6,120)(  7,119)(  8,118)(  9,117)
( 10,116)( 11,115)( 12,114)( 13,113)( 14,112)( 15,111)( 16,110)( 17,109)
( 18,108)( 19,107)( 20,140)( 21,139)( 22,155)( 23,154)( 24,153)( 25,152)
( 26,151)( 27,150)( 28,149)( 29,148)( 30,147)( 31,146)( 32,145)( 33,144)
( 34,143)( 35,142)( 36,141)( 37,123)( 38,122)( 39,138)( 40,137)( 41,136)
( 42,135)( 43,134)( 44,133)( 45,132)( 46,131)( 47,130)( 48,129)( 49,128)
( 50,127)( 51,126)( 52,125)( 53,124)( 54,157)( 55,156)( 56,172)( 57,171)
( 58,170)( 59,169)( 60,168)( 61,167)( 62,166)( 63,165)( 64,164)( 65,163)
( 66,162)( 67,161)( 68,160)( 69,159)( 70,158)( 71,191)( 72,190)( 73,206)
( 74,205)( 75,204)( 76,203)( 77,202)( 78,201)( 79,200)( 80,199)( 81,198)
( 82,197)( 83,196)( 84,195)( 85,194)( 86,193)( 87,192)( 88,174)( 89,173)
( 90,189)( 91,188)( 92,187)( 93,186)( 94,185)( 95,184)( 96,183)( 97,182)
( 98,181)( 99,180)(100,179)(101,178)(102,177)(103,176)(104,175);
s3 := Sym(206)!(  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)
( 18, 35)( 19, 36)( 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)( 69, 86)( 70, 87)(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)(120,137)(121,138)(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)(171,188)(172,189);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, 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*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