Polytope of Type {22,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {22,4,4}*1408
Also Known As : {{22,4|2},{4,4}₄}. if this polytope has another name.
Group : SmallGroup(1408,13892)
Rank : 4
Schlafli Type : {22,4,4}
Number of vertices, edges, etc : 22, 88, 16, 8
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 : {22,4,4}*704
   4-fold quotients : {22,2,4}*352, {22,4,2}*352
   8-fold quotients : {11,2,4}*176, {22,2,2}*176
   11-fold quotients : {2,4,4}*128
   16-fold quotients : {11,2,2}*88
   22-fold quotients : {2,4,4}*64
   44-fold quotients : {2,2,4}*32, {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, 45)(  2, 55)(  3, 54)(  4, 53)(  5, 52)(  6, 51)(  7, 50)(  8, 49)
(  9, 48)( 10, 47)( 11, 46)( 12, 56)( 13, 66)( 14, 65)( 15, 64)( 16, 63)
( 17, 62)( 18, 61)( 19, 60)( 20, 59)( 21, 58)( 22, 57)( 23, 67)( 24, 77)
( 25, 76)( 26, 75)( 27, 74)( 28, 73)( 29, 72)( 30, 71)( 31, 70)( 32, 69)
( 33, 68)( 34, 78)( 35, 88)( 36, 87)( 37, 86)( 38, 85)( 39, 84)( 40, 83)
( 41, 82)( 42, 81)( 43, 80)( 44, 79)( 89,133)( 90,143)( 91,142)( 92,141)
( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,134)(100,144)
(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)
(109,146)(110,145)(111,155)(112,165)(113,164)(114,163)(115,162)(116,161)
(117,160)(118,159)(119,158)(120,157)(121,156)(122,166)(123,176)(124,175)
(125,174)(126,173)(127,172)(128,171)(129,170)(130,169)(131,168)(132,167);;
s1 := (  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, 79)( 24, 78)
( 25, 88)( 26, 87)( 27, 86)( 28, 85)( 29, 84)( 30, 83)( 31, 82)( 32, 81)
( 33, 80)( 34, 68)( 35, 67)( 36, 77)( 37, 76)( 38, 75)( 39, 74)( 40, 73)
( 41, 72)( 42, 71)( 43, 70)( 44, 69)( 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)(111,167)(112,166)(113,176)(114,175)(115,174)(116,173)
(117,172)(118,171)(119,170)(120,169)(121,168)(122,156)(123,155)(124,165)
(125,164)(126,163)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157);;
s2 := ( 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);;
s3 := (  1, 89)(  2, 90)(  3, 91)(  4, 92)(  5, 93)(  6, 94)(  7, 95)(  8, 96)
(  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)( 16,104)
( 17,105)( 18,106)( 19,107)( 20,108)( 21,109)( 22,110)( 23,111)( 24,112)
( 25,113)( 26,114)( 27,115)( 28,116)( 29,117)( 30,118)( 31,119)( 32,120)
( 33,121)( 34,122)( 35,123)( 36,124)( 37,125)( 38,126)( 39,127)( 40,128)
( 41,129)( 42,130)( 43,131)( 44,132)( 45,133)( 46,134)( 47,135)( 48,136)
( 49,137)( 50,138)( 51,139)( 52,140)( 53,141)( 54,142)( 55,143)( 56,144)
( 57,145)( 58,146)( 59,147)( 60,148)( 61,149)( 62,150)( 63,151)( 64,152)
( 65,153)( 66,154)( 67,155)( 68,156)( 69,157)( 70,158)( 71,159)( 72,160)
( 73,161)( 74,162)( 75,163)( 76,164)( 77,165)( 78,166)( 79,167)( 80,168)
( 81,169)( 82,170)( 83,171)( 84,172)( 85,173)( 86,174)( 87,175)( 88,176);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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