Polytope of Type {6,66,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,66,2}*1584b
if this polytope has a name.
Group : SmallGroup(1584,688)
Rank : 4
Schlafli Type : {6,66,2}
Number of vertices, edges, etc : 6, 198, 66, 2
Order of s₀s₁s₂s₃ : 66
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) :
   3-fold quotients : {6,22,2}*528, {2,66,2}*528
   6-fold quotients : {2,33,2}*264
   9-fold quotients : {2,22,2}*176
   11-fold quotients : {6,6,2}*144a
   18-fold quotients : {2,11,2}*88
   33-fold quotients : {2,6,2}*48, {6,2,2}*48
   66-fold quotients : {2,3,2}*24, {3,2,2}*24
   99-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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