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Polytope of Type {2,2,21,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,21,4}*1344
if this polytope has a name.
Group : SmallGroup(1344,11701)
Rank : 5
Schlafli Type : {2,2,21,4}
Number of vertices, edges, etc : 2, 2, 42, 84, 8
Order of s0s1s2s3s4 : 42
Order of s0s1s2s3s4s3s2s1 : 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 : {2,2,21,4}*672
   4-fold quotients : {2,2,21,2}*336
   7-fold quotients : {2,2,3,4}*192
   12-fold quotients : {2,2,7,2}*112
   14-fold quotients : {2,2,3,4}*96
   28-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := (  6,  7)(  9, 29)( 10, 31)( 11, 30)( 12, 32)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 33, 61)( 34, 63)( 35, 62)
( 36, 64)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 81)( 42, 83)( 43, 82)
( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)( 59, 66)
( 60, 68)( 90, 91)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)( 98,111)
( 99,110)(100,112)(101,105)(102,107)(103,106)(104,108)(117,145)(118,147)
(119,146)(120,148)(121,169)(122,171)(123,170)(124,172)(125,165)(126,167)
(127,166)(128,168)(129,161)(130,163)(131,162)(132,164)(133,157)(134,159)
(135,158)(136,160)(137,153)(138,155)(139,154)(140,156)(141,149)(142,151)
(143,150)(144,152);;
s3 := (  5, 37)(  6, 38)(  7, 40)(  8, 39)(  9, 33)( 10, 34)( 11, 36)( 12, 35)
( 13, 57)( 14, 58)( 15, 60)( 16, 59)( 17, 53)( 18, 54)( 19, 56)( 20, 55)
( 21, 49)( 22, 50)( 23, 52)( 24, 51)( 25, 45)( 26, 46)( 27, 48)( 28, 47)
( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 61, 65)( 62, 66)( 63, 68)( 64, 67)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 81)( 74, 82)( 75, 84)( 76, 83)
( 79, 80)( 89,121)( 90,122)( 91,124)( 92,123)( 93,117)( 94,118)( 95,120)
( 96,119)( 97,141)( 98,142)( 99,144)(100,143)(101,137)(102,138)(103,140)
(104,139)(105,133)(106,134)(107,136)(108,135)(109,129)(110,130)(111,132)
(112,131)(113,125)(114,126)(115,128)(116,127)(145,149)(146,150)(147,152)
(148,151)(153,169)(154,170)(155,172)(156,171)(157,165)(158,166)(159,168)
(160,167)(163,164);;
s4 := (  5, 92)(  6, 91)(  7, 90)(  8, 89)(  9, 96)( 10, 95)( 11, 94)( 12, 93)
( 13,100)( 14, 99)( 15, 98)( 16, 97)( 17,104)( 18,103)( 19,102)( 20,101)
( 21,108)( 22,107)( 23,106)( 24,105)( 25,112)( 26,111)( 27,110)( 28,109)
( 29,116)( 30,115)( 31,114)( 32,113)( 33,120)( 34,119)( 35,118)( 36,117)
( 37,124)( 38,123)( 39,122)( 40,121)( 41,128)( 42,127)( 43,126)( 44,125)
( 45,132)( 46,131)( 47,130)( 48,129)( 49,136)( 50,135)( 51,134)( 52,133)
( 53,140)( 54,139)( 55,138)( 56,137)( 57,144)( 58,143)( 59,142)( 60,141)
( 61,148)( 62,147)( 63,146)( 64,145)( 65,152)( 66,151)( 67,150)( 68,149)
( 69,156)( 70,155)( 71,154)( 72,153)( 73,160)( 74,159)( 75,158)( 76,157)
( 77,164)( 78,163)( 79,162)( 80,161)( 81,168)( 82,167)( 83,166)( 84,165)
( 85,172)( 86,171)( 87,170)( 88,169);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*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*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(172)!(1,2);
s1 := Sym(172)!(3,4);
s2 := Sym(172)!(  6,  7)(  9, 29)( 10, 31)( 11, 30)( 12, 32)( 13, 25)( 14, 27)
( 15, 26)( 16, 28)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 33, 61)( 34, 63)
( 35, 62)( 36, 64)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 81)( 42, 83)
( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 73)( 50, 75)
( 51, 74)( 52, 76)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)
( 59, 66)( 60, 68)( 90, 91)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)
( 98,111)( 99,110)(100,112)(101,105)(102,107)(103,106)(104,108)(117,145)
(118,147)(119,146)(120,148)(121,169)(122,171)(123,170)(124,172)(125,165)
(126,167)(127,166)(128,168)(129,161)(130,163)(131,162)(132,164)(133,157)
(134,159)(135,158)(136,160)(137,153)(138,155)(139,154)(140,156)(141,149)
(142,151)(143,150)(144,152);
s3 := Sym(172)!(  5, 37)(  6, 38)(  7, 40)(  8, 39)(  9, 33)( 10, 34)( 11, 36)
( 12, 35)( 13, 57)( 14, 58)( 15, 60)( 16, 59)( 17, 53)( 18, 54)( 19, 56)
( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)( 25, 45)( 26, 46)( 27, 48)
( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 61, 65)( 62, 66)( 63, 68)
( 64, 67)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 81)( 74, 82)( 75, 84)
( 76, 83)( 79, 80)( 89,121)( 90,122)( 91,124)( 92,123)( 93,117)( 94,118)
( 95,120)( 96,119)( 97,141)( 98,142)( 99,144)(100,143)(101,137)(102,138)
(103,140)(104,139)(105,133)(106,134)(107,136)(108,135)(109,129)(110,130)
(111,132)(112,131)(113,125)(114,126)(115,128)(116,127)(145,149)(146,150)
(147,152)(148,151)(153,169)(154,170)(155,172)(156,171)(157,165)(158,166)
(159,168)(160,167)(163,164);
s4 := Sym(172)!(  5, 92)(  6, 91)(  7, 90)(  8, 89)(  9, 96)( 10, 95)( 11, 94)
( 12, 93)( 13,100)( 14, 99)( 15, 98)( 16, 97)( 17,104)( 18,103)( 19,102)
( 20,101)( 21,108)( 22,107)( 23,106)( 24,105)( 25,112)( 26,111)( 27,110)
( 28,109)( 29,116)( 30,115)( 31,114)( 32,113)( 33,120)( 34,119)( 35,118)
( 36,117)( 37,124)( 38,123)( 39,122)( 40,121)( 41,128)( 42,127)( 43,126)
( 44,125)( 45,132)( 46,131)( 47,130)( 48,129)( 49,136)( 50,135)( 51,134)
( 52,133)( 53,140)( 54,139)( 55,138)( 56,137)( 57,144)( 58,143)( 59,142)
( 60,141)( 61,148)( 62,147)( 63,146)( 64,145)( 65,152)( 66,151)( 67,150)
( 68,149)( 69,156)( 70,155)( 71,154)( 72,153)( 73,160)( 74,159)( 75,158)
( 76,157)( 77,164)( 78,163)( 79,162)( 80,161)( 81,168)( 82,167)( 83,166)
( 84,165)( 85,172)( 86,171)( 87,170)( 88,169);
poly := sub<Sym(172)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4*s3*s4, s2*s3*s4*s3*s2*s3*s2*s3*s4*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*s2*s3 >; 
 

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