Polytope of Type {4,22,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,22,4}*704
Also Known As : {{4,22|2},{22,4|2}}. if this polytope has another name.
Group : SmallGroup(704,1040)
Rank : 4
Schlafli Type : {4,22,4}
Number of vertices, edges, etc : 4, 44, 44, 4
Order of s₀s₁s₂s₃ : 44
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,22,4,2} of size 1408
Vertex Figure Of :
   {2,4,22,4} of size 1408
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,22,4}*352, {4,22,2}*352
   4-fold quotients : {2,22,2}*176
   8-fold quotients : {2,11,2}*88
   11-fold quotients : {4,2,4}*64
   22-fold quotients : {2,2,4}*32, {4,2,2}*32
   44-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,44,4}*1408, {4,22,8}*1408, {8,22,4}*1408
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope