Polytope of Type {16,22,2}

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

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

Permutation Representation (Magma) :

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

to this polytope