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# Polytope of Type {2,66,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,66,6}*1584b
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) :
3-fold quotients : {2,22,6}*528, {2,66,2}*528
6-fold quotients : {2,33,2}*264
9-fold quotients : {2,22,2}*176
11-fold quotients : {2,6,6}*144a
18-fold quotients : {2,11,2}*88
33-fold quotients : {2,2,6}*48, {2,6,2}*48
66-fold quotients : {2,2,3}*24, {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)
( 37, 46)( 38, 45)( 39, 44)( 40, 43)( 41, 42)( 47, 58)( 48, 68)( 49, 67)
( 50, 66)( 51, 65)( 52, 64)( 53, 63)( 54, 62)( 55, 61)( 56, 60)( 57, 59)
( 70, 79)( 71, 78)( 72, 77)( 73, 76)( 74, 75)( 80, 91)( 81,101)( 82,100)
( 83, 99)( 84, 98)( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)( 90, 92)
(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)
(136,145)(137,144)(138,143)(139,142)(140,141)(146,157)(147,167)(148,166)
(149,165)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)(156,158)
(169,178)(170,177)(171,176)(172,175)(173,174)(179,190)(180,200)(181,199)
(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191);;
s2 := (  3, 15)(  4, 14)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)( 10, 19)
( 11, 18)( 12, 17)( 13, 16)( 25, 26)( 27, 35)( 28, 34)( 29, 33)( 30, 32)
( 36, 81)( 37, 80)( 38, 90)( 39, 89)( 40, 88)( 41, 87)( 42, 86)( 43, 85)
( 44, 84)( 45, 83)( 46, 82)( 47, 70)( 48, 69)( 49, 79)( 50, 78)( 51, 77)
( 52, 76)( 53, 75)( 54, 74)( 55, 73)( 56, 72)( 57, 71)( 58, 92)( 59, 91)
( 60,101)( 61,100)( 62, 99)( 63, 98)( 64, 97)( 65, 96)( 66, 95)( 67, 94)
( 68, 93)(102,114)(103,113)(104,123)(105,122)(106,121)(107,120)(108,119)
(109,118)(110,117)(111,116)(112,115)(124,125)(126,134)(127,133)(128,132)
(129,131)(135,180)(136,179)(137,189)(138,188)(139,187)(140,186)(141,185)
(142,184)(143,183)(144,182)(145,181)(146,169)(147,168)(148,178)(149,177)
(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)(157,191)
(158,190)(159,200)(160,199)(161,198)(162,197)(163,196)(164,195)(165,194)
(166,193)(167,192);;
s3 := (  3,135)(  4,136)(  5,137)(  6,138)(  7,139)(  8,140)(  9,141)( 10,142)
( 11,143)( 12,144)( 13,145)( 14,146)( 15,147)( 16,148)( 17,149)( 18,150)
( 19,151)( 20,152)( 21,153)( 22,154)( 23,155)( 24,156)( 25,157)( 26,158)
( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)( 32,164)( 33,165)( 34,166)
( 35,167)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)( 41,107)( 42,108)
( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)( 48,114)( 49,115)( 50,116)
( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)( 56,122)( 57,123)( 58,124)
( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)( 65,131)( 66,132)
( 67,133)( 68,134)( 69,168)( 70,169)( 71,170)( 72,171)( 73,172)( 74,173)
( 75,174)( 76,175)( 77,176)( 78,177)( 79,178)( 80,179)( 81,180)( 82,181)
( 83,182)( 84,183)( 85,184)( 86,185)( 87,186)( 88,187)( 89,188)( 90,189)
( 91,190)( 92,191)( 93,192)( 94,193)( 95,194)( 96,195)( 97,196)( 98,197)
( 99,198)(100,199)(101,200);;
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,
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 ];;
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)( 37, 46)( 38, 45)( 39, 44)( 40, 43)( 41, 42)( 47, 58)( 48, 68)
( 49, 67)( 50, 66)( 51, 65)( 52, 64)( 53, 63)( 54, 62)( 55, 61)( 56, 60)
( 57, 59)( 70, 79)( 71, 78)( 72, 77)( 73, 76)( 74, 75)( 80, 91)( 81,101)
( 82,100)( 83, 99)( 84, 98)( 85, 97)( 86, 96)( 87, 95)( 88, 94)( 89, 93)
( 90, 92)(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)(136,145)(137,144)(138,143)(139,142)(140,141)(146,157)(147,167)
(148,166)(149,165)(150,164)(151,163)(152,162)(153,161)(154,160)(155,159)
(156,158)(169,178)(170,177)(171,176)(172,175)(173,174)(179,190)(180,200)
(181,199)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)
(189,191);
s2 := Sym(200)!(  3, 15)(  4, 14)(  5, 24)(  6, 23)(  7, 22)(  8, 21)(  9, 20)
( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 25, 26)( 27, 35)( 28, 34)( 29, 33)
( 30, 32)( 36, 81)( 37, 80)( 38, 90)( 39, 89)( 40, 88)( 41, 87)( 42, 86)
( 43, 85)( 44, 84)( 45, 83)( 46, 82)( 47, 70)( 48, 69)( 49, 79)( 50, 78)
( 51, 77)( 52, 76)( 53, 75)( 54, 74)( 55, 73)( 56, 72)( 57, 71)( 58, 92)
( 59, 91)( 60,101)( 61,100)( 62, 99)( 63, 98)( 64, 97)( 65, 96)( 66, 95)
( 67, 94)( 68, 93)(102,114)(103,113)(104,123)(105,122)(106,121)(107,120)
(108,119)(109,118)(110,117)(111,116)(112,115)(124,125)(126,134)(127,133)
(128,132)(129,131)(135,180)(136,179)(137,189)(138,188)(139,187)(140,186)
(141,185)(142,184)(143,183)(144,182)(145,181)(146,169)(147,168)(148,178)
(149,177)(150,176)(151,175)(152,174)(153,173)(154,172)(155,171)(156,170)
(157,191)(158,190)(159,200)(160,199)(161,198)(162,197)(163,196)(164,195)
(165,194)(166,193)(167,192);
s3 := Sym(200)!(  3,135)(  4,136)(  5,137)(  6,138)(  7,139)(  8,140)(  9,141)
( 10,142)( 11,143)( 12,144)( 13,145)( 14,146)( 15,147)( 16,148)( 17,149)
( 18,150)( 19,151)( 20,152)( 21,153)( 22,154)( 23,155)( 24,156)( 25,157)
( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)( 32,164)( 33,165)
( 34,166)( 35,167)( 36,102)( 37,103)( 38,104)( 39,105)( 40,106)( 41,107)
( 42,108)( 43,109)( 44,110)( 45,111)( 46,112)( 47,113)( 48,114)( 49,115)
( 50,116)( 51,117)( 52,118)( 53,119)( 54,120)( 55,121)( 56,122)( 57,123)
( 58,124)( 59,125)( 60,126)( 61,127)( 62,128)( 63,129)( 64,130)( 65,131)
( 66,132)( 67,133)( 68,134)( 69,168)( 70,169)( 71,170)( 72,171)( 73,172)
( 74,173)( 75,174)( 76,175)( 77,176)( 78,177)( 79,178)( 80,179)( 81,180)
( 82,181)( 83,182)( 84,183)( 85,184)( 86,185)( 87,186)( 88,187)( 89,188)
( 90,189)( 91,190)( 92,191)( 93,192)( 94,193)( 95,194)( 96,195)( 97,196)
( 98,197)( 99,198)(100,199)(101,200);
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, 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 >;

```

to this polytope