Polytope of Type {2,66,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,66,6}*1584c
if this polytope has a name.
Group : SmallGroup(1584,688)
Rank : 4
Schlafli Type : {2,66,6}
Number of vertices, edges, etc : 2, 66, 198, 6
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) :
   2-fold quotients : {2,33,6}*792
   3-fold quotients : {2,66,2}*528
   6-fold quotients : {2,33,2}*264
   9-fold quotients : {2,22,2}*176
   11-fold quotients : {2,6,6}*144c
   18-fold quotients : {2,11,2}*88
   22-fold quotients : {2,3,6}*72
   33-fold quotients : {2,6,2}*48
   66-fold quotients : {2,3,2}*24
   99-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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