Polytope of Type {2,16,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,16,22}*1408
if this polytope has a name.
Group : SmallGroup(1408,17614)
Rank : 4
Schlafli Type : {2,16,22}
Number of vertices, edges, etc : 2, 16, 176, 22
Order of s₀s₁s₂s₃ : 176
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 : {2,8,22}*704
   4-fold quotients : {2,4,22}*352
   8-fold quotients : {2,2,22}*176
   11-fold quotients : {2,16,2}*128
   16-fold quotients : {2,2,11}*88
   22-fold quotients : {2,8,2}*64
   44-fold quotients : {2,4,2}*32
   88-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope