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Polytope of Type {4,4,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,22}*1408
Also Known As : {{4,4}4,{4,22|2}}. if this polytope has another name.
Group : SmallGroup(1408,13892)
Rank : 4
Schlafli Type : {4,4,22}
Number of vertices, edges, etc : 8, 16, 88, 22
Order of s0s1s2s3 : 44
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {4,4,22}*704
   4-fold quotients : {2,4,22}*352, {4,2,22}*352
   8-fold quotients : {4,2,11}*176, {2,2,22}*176
   11-fold quotients : {4,4,2}*128
   16-fold quotients : {2,2,11}*88
   22-fold quotients : {4,4,2}*64
   44-fold quotients : {2,4,2}*32, {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)( 67, 78)( 68, 79)( 69, 80)( 70, 81)( 71, 82)
( 72, 83)( 73, 84)( 74, 85)( 75, 86)( 76, 87)( 77, 88)(111,122)(112,123)
(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)
(121,132)(155,166)(156,167)(157,168)(158,169)(159,170)(160,171)(161,172)
(162,173)(163,174)(164,175)(165,176);;
s1 := ( 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);;
s2 := (  1,133)(  2,143)(  3,142)(  4,141)(  5,140)(  6,139)(  7,138)(  8,137)
(  9,136)( 10,135)( 11,134)( 12,144)( 13,154)( 14,153)( 15,152)( 16,151)
( 17,150)( 18,149)( 19,148)( 20,147)( 21,146)( 22,145)( 23,155)( 24,165)
( 25,164)( 26,163)( 27,162)( 28,161)( 29,160)( 30,159)( 31,158)( 32,157)
( 33,156)( 34,166)( 35,176)( 36,175)( 37,174)( 38,173)( 39,172)( 40,171)
( 41,170)( 42,169)( 43,168)( 44,167)( 45, 89)( 46, 99)( 47, 98)( 48, 97)
( 49, 96)( 50, 95)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 90)( 56,100)
( 57,110)( 58,109)( 59,108)( 60,107)( 61,106)( 62,105)( 63,104)( 64,103)
( 65,102)( 66,101)( 67,111)( 68,121)( 69,120)( 70,119)( 71,118)( 72,117)
( 73,116)( 74,115)( 75,114)( 76,113)( 77,112)( 78,122)( 79,132)( 80,131)
( 81,130)( 82,129)( 83,128)( 84,127)( 85,126)( 86,125)( 87,124)( 88,123);;
s3 := (  1, 46)(  2, 45)(  3, 55)(  4, 54)(  5, 53)(  6, 52)(  7, 51)(  8, 50)
(  9, 49)( 10, 48)( 11, 47)( 12, 57)( 13, 56)( 14, 66)( 15, 65)( 16, 64)
( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 68)( 24, 67)
( 25, 77)( 26, 76)( 27, 75)( 28, 74)( 29, 73)( 30, 72)( 31, 71)( 32, 70)
( 33, 69)( 34, 79)( 35, 78)( 36, 88)( 37, 87)( 38, 86)( 39, 85)( 40, 84)
( 41, 83)( 42, 82)( 43, 81)( 44, 80)( 89,134)( 90,133)( 91,143)( 92,142)
( 93,141)( 94,140)( 95,139)( 96,138)( 97,137)( 98,136)( 99,135)(100,145)
(101,144)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)(108,148)
(109,147)(110,146)(111,156)(112,155)(113,165)(114,164)(115,163)(116,162)
(117,161)(118,160)(119,159)(120,158)(121,157)(122,167)(123,166)(124,176)
(125,175)(126,174)(127,173)(128,172)(129,171)(130,170)(131,169)(132,168);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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(176)!( 23, 34)( 24, 35)( 25, 36)( 26, 37)( 27, 38)( 28, 39)( 29, 40)
( 30, 41)( 31, 42)( 32, 43)( 33, 44)( 67, 78)( 68, 79)( 69, 80)( 70, 81)
( 71, 82)( 72, 83)( 73, 84)( 74, 85)( 75, 86)( 76, 87)( 77, 88)(111,122)
(112,123)(113,124)(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)
(120,131)(121,132)(155,166)(156,167)(157,168)(158,169)(159,170)(160,171)
(161,172)(162,173)(163,174)(164,175)(165,176);
s1 := 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);
s2 := Sym(176)!(  1,133)(  2,143)(  3,142)(  4,141)(  5,140)(  6,139)(  7,138)
(  8,137)(  9,136)( 10,135)( 11,134)( 12,144)( 13,154)( 14,153)( 15,152)
( 16,151)( 17,150)( 18,149)( 19,148)( 20,147)( 21,146)( 22,145)( 23,155)
( 24,165)( 25,164)( 26,163)( 27,162)( 28,161)( 29,160)( 30,159)( 31,158)
( 32,157)( 33,156)( 34,166)( 35,176)( 36,175)( 37,174)( 38,173)( 39,172)
( 40,171)( 41,170)( 42,169)( 43,168)( 44,167)( 45, 89)( 46, 99)( 47, 98)
( 48, 97)( 49, 96)( 50, 95)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 90)
( 56,100)( 57,110)( 58,109)( 59,108)( 60,107)( 61,106)( 62,105)( 63,104)
( 64,103)( 65,102)( 66,101)( 67,111)( 68,121)( 69,120)( 70,119)( 71,118)
( 72,117)( 73,116)( 74,115)( 75,114)( 76,113)( 77,112)( 78,122)( 79,132)
( 80,131)( 81,130)( 82,129)( 83,128)( 84,127)( 85,126)( 86,125)( 87,124)
( 88,123);
s3 := Sym(176)!(  1, 46)(  2, 45)(  3, 55)(  4, 54)(  5, 53)(  6, 52)(  7, 51)
(  8, 50)(  9, 49)( 10, 48)( 11, 47)( 12, 57)( 13, 56)( 14, 66)( 15, 65)
( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 68)
( 24, 67)( 25, 77)( 26, 76)( 27, 75)( 28, 74)( 29, 73)( 30, 72)( 31, 71)
( 32, 70)( 33, 69)( 34, 79)( 35, 78)( 36, 88)( 37, 87)( 38, 86)( 39, 85)
( 40, 84)( 41, 83)( 42, 82)( 43, 81)( 44, 80)( 89,134)( 90,133)( 91,143)
( 92,142)( 93,141)( 94,140)( 95,139)( 96,138)( 97,137)( 98,136)( 99,135)
(100,145)(101,144)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)
(108,148)(109,147)(110,146)(111,156)(112,155)(113,165)(114,164)(115,163)
(116,162)(117,161)(118,160)(119,159)(120,158)(121,157)(122,167)(123,166)
(124,176)(125,175)(126,174)(127,173)(128,172)(129,171)(130,170)(131,169)
(132,168);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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 >; 
 
References : None.
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