Polytope of Type {2,16,26}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,16,26}*1664
if this polytope has a name.
Group : SmallGroup(1664,17614)
Rank : 4
Schlafli Type : {2,16,26}
Number of vertices, edges, etc : 2, 16, 208, 26
Order of s₀s₁s₂s₃ : 208
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 : {2,8,26}*832
   4-fold quotients : {2,4,26}*416
   8-fold quotients : {2,2,26}*208
   13-fold quotients : {2,16,2}*128
   16-fold quotients : {2,2,13}*104
   26-fold quotients : {2,8,2}*64
   52-fold quotients : {2,4,2}*32
   104-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 29, 42)( 30, 43)( 31, 44)( 32, 45)( 33, 46)( 34, 47)( 35, 48)( 36, 49)
( 37, 50)( 38, 51)( 39, 52)( 40, 53)( 41, 54)( 55, 81)( 56, 82)( 57, 83)
( 58, 84)( 59, 85)( 60, 86)( 61, 87)( 62, 88)( 63, 89)( 64, 90)( 65, 91)
( 66, 92)( 67, 93)( 68, 94)( 69, 95)( 70, 96)( 71, 97)( 72, 98)( 73, 99)
( 74,100)( 75,101)( 76,102)( 77,103)( 78,104)( 79,105)( 80,106)(107,159)
(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)(115,167)
(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)(123,175)
(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)(131,183)
(132,184)(133,198)(134,199)(135,200)(136,201)(137,202)(138,203)(139,204)
(140,205)(141,206)(142,207)(143,208)(144,209)(145,210)(146,185)(147,186)
(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)
(156,195)(157,196)(158,197);;
s2 := (  3,107)(  4,119)(  5,118)(  6,117)(  7,116)(  8,115)(  9,114)( 10,113)
( 11,112)( 12,111)( 13,110)( 14,109)( 15,108)( 16,120)( 17,132)( 18,131)
( 19,130)( 20,129)( 21,128)( 22,127)( 23,126)( 24,125)( 25,124)( 26,123)
( 27,122)( 28,121)( 29,146)( 30,158)( 31,157)( 32,156)( 33,155)( 34,154)
( 35,153)( 36,152)( 37,151)( 38,150)( 39,149)( 40,148)( 41,147)( 42,133)
( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,140)( 49,139)( 50,138)
( 51,137)( 52,136)( 53,135)( 54,134)( 55,185)( 56,197)( 57,196)( 58,195)
( 59,194)( 60,193)( 61,192)( 62,191)( 63,190)( 64,189)( 65,188)( 66,187)
( 67,186)( 68,198)( 69,210)( 70,209)( 71,208)( 72,207)( 73,206)( 74,205)
( 75,204)( 76,203)( 77,202)( 78,201)( 79,200)( 80,199)( 81,159)( 82,171)
( 83,170)( 84,169)( 85,168)( 86,167)( 87,166)( 88,165)( 89,164)( 90,163)
( 91,162)( 92,161)( 93,160)( 94,172)( 95,184)( 96,183)( 97,182)( 98,181)
( 99,180)(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)(106,173);;
s3 := (  3,  4)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)( 16, 17)( 18, 28)
( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 29, 30)( 31, 41)( 32, 40)( 33, 39)
( 34, 38)( 35, 37)( 42, 43)( 44, 54)( 45, 53)( 46, 52)( 47, 51)( 48, 50)
( 55, 56)( 57, 67)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 68, 69)( 70, 80)
( 71, 79)( 72, 78)( 73, 77)( 74, 76)( 81, 82)( 83, 93)( 84, 92)( 85, 91)
( 86, 90)( 87, 89)( 94, 95)( 96,106)( 97,105)( 98,104)( 99,103)(100,102)
(107,108)(109,119)(110,118)(111,117)(112,116)(113,115)(120,121)(122,132)
(123,131)(124,130)(125,129)(126,128)(133,134)(135,145)(136,144)(137,143)
(138,142)(139,141)(146,147)(148,158)(149,157)(150,156)(151,155)(152,154)
(159,160)(161,171)(162,170)(163,169)(164,168)(165,167)(172,173)(174,184)
(175,183)(176,182)(177,181)(178,180)(185,186)(187,197)(188,196)(189,195)
(190,194)(191,193)(198,199)(200,210)(201,209)(202,208)(203,207)(204,206);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(210)!(1,2);
s1 := Sym(210)!( 29, 42)( 30, 43)( 31, 44)( 32, 45)( 33, 46)( 34, 47)( 35, 48)
( 36, 49)( 37, 50)( 38, 51)( 39, 52)( 40, 53)( 41, 54)( 55, 81)( 56, 82)
( 57, 83)( 58, 84)( 59, 85)( 60, 86)( 61, 87)( 62, 88)( 63, 89)( 64, 90)
( 65, 91)( 66, 92)( 67, 93)( 68, 94)( 69, 95)( 70, 96)( 71, 97)( 72, 98)
( 73, 99)( 74,100)( 75,101)( 76,102)( 77,103)( 78,104)( 79,105)( 80,106)
(107,159)(108,160)(109,161)(110,162)(111,163)(112,164)(113,165)(114,166)
(115,167)(116,168)(117,169)(118,170)(119,171)(120,172)(121,173)(122,174)
(123,175)(124,176)(125,177)(126,178)(127,179)(128,180)(129,181)(130,182)
(131,183)(132,184)(133,198)(134,199)(135,200)(136,201)(137,202)(138,203)
(139,204)(140,205)(141,206)(142,207)(143,208)(144,209)(145,210)(146,185)
(147,186)(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)
(155,194)(156,195)(157,196)(158,197);
s2 := Sym(210)!(  3,107)(  4,119)(  5,118)(  6,117)(  7,116)(  8,115)(  9,114)
( 10,113)( 11,112)( 12,111)( 13,110)( 14,109)( 15,108)( 16,120)( 17,132)
( 18,131)( 19,130)( 20,129)( 21,128)( 22,127)( 23,126)( 24,125)( 25,124)
( 26,123)( 27,122)( 28,121)( 29,146)( 30,158)( 31,157)( 32,156)( 33,155)
( 34,154)( 35,153)( 36,152)( 37,151)( 38,150)( 39,149)( 40,148)( 41,147)
( 42,133)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,140)( 49,139)
( 50,138)( 51,137)( 52,136)( 53,135)( 54,134)( 55,185)( 56,197)( 57,196)
( 58,195)( 59,194)( 60,193)( 61,192)( 62,191)( 63,190)( 64,189)( 65,188)
( 66,187)( 67,186)( 68,198)( 69,210)( 70,209)( 71,208)( 72,207)( 73,206)
( 74,205)( 75,204)( 76,203)( 77,202)( 78,201)( 79,200)( 80,199)( 81,159)
( 82,171)( 83,170)( 84,169)( 85,168)( 86,167)( 87,166)( 88,165)( 89,164)
( 90,163)( 91,162)( 92,161)( 93,160)( 94,172)( 95,184)( 96,183)( 97,182)
( 98,181)( 99,180)(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)
(106,173);
s3 := Sym(210)!(  3,  4)(  5, 15)(  6, 14)(  7, 13)(  8, 12)(  9, 11)( 16, 17)
( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 29, 30)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 43)( 44, 54)( 45, 53)( 46, 52)( 47, 51)
( 48, 50)( 55, 56)( 57, 67)( 58, 66)( 59, 65)( 60, 64)( 61, 63)( 68, 69)
( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)( 81, 82)( 83, 93)( 84, 92)
( 85, 91)( 86, 90)( 87, 89)( 94, 95)( 96,106)( 97,105)( 98,104)( 99,103)
(100,102)(107,108)(109,119)(110,118)(111,117)(112,116)(113,115)(120,121)
(122,132)(123,131)(124,130)(125,129)(126,128)(133,134)(135,145)(136,144)
(137,143)(138,142)(139,141)(146,147)(148,158)(149,157)(150,156)(151,155)
(152,154)(159,160)(161,171)(162,170)(163,169)(164,168)(165,167)(172,173)
(174,184)(175,183)(176,182)(177,181)(178,180)(185,186)(187,197)(188,196)
(189,195)(190,194)(191,193)(198,199)(200,210)(201,209)(202,208)(203,207)
(204,206);
poly := sub<Sym(210)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope