Polytope of Type {16,26,2}

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

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

Permutation Representation (Magma) :

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

to this polytope