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Polytope of Type {168,2,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {168,2,2}*1344
if this polytope has a name.
Group : SmallGroup(1344,11119)
Rank : 4
Schlafli Type : {168,2,2}
Number of vertices, edges, etc : 168, 168, 2, 2
Order of s0s1s2s3 : 168
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 : {84,2,2}*672
   3-fold quotients : {56,2,2}*448
   4-fold quotients : {42,2,2}*336
   6-fold quotients : {28,2,2}*224
   7-fold quotients : {24,2,2}*192
   8-fold quotients : {21,2,2}*168
   12-fold quotients : {14,2,2}*112
   14-fold quotients : {12,2,2}*96
   21-fold quotients : {8,2,2}*64
   24-fold quotients : {7,2,2}*56
   28-fold quotients : {6,2,2}*48
   42-fold quotients : {4,2,2}*32
   56-fold quotients : {3,2,2}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)
( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 64)( 44, 70)( 45, 69)( 46, 68)
( 47, 67)( 48, 66)( 49, 65)( 50, 78)( 51, 84)( 52, 83)( 53, 82)( 54, 81)
( 55, 80)( 56, 79)( 57, 71)( 58, 77)( 59, 76)( 60, 75)( 61, 74)( 62, 73)
( 63, 72)( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)
( 92,141)( 93,147)( 94,146)( 95,145)( 96,144)( 97,143)( 98,142)( 99,134)
(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,148)(107,154)
(108,153)(109,152)(110,151)(111,150)(112,149)(113,162)(114,168)(115,167)
(116,166)(117,165)(118,164)(119,163)(120,155)(121,161)(122,160)(123,159)
(124,158)(125,157)(126,156);;
s1 := (  1, 93)(  2, 92)(  3, 98)(  4, 97)(  5, 96)(  6, 95)(  7, 94)(  8, 86)
(  9, 85)( 10, 91)( 11, 90)( 12, 89)( 13, 88)( 14, 87)( 15,100)( 16, 99)
( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,114)( 23,113)( 24,119)
( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)( 32,111)
( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)( 40,124)
( 41,123)( 42,122)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,149)( 51,148)( 52,154)( 53,153)( 54,152)( 55,151)( 56,150)
( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,135)
( 65,134)( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)( 72,127)
( 73,133)( 74,132)( 75,131)( 76,130)( 77,129)( 78,142)( 79,141)( 80,147)
( 81,146)( 82,145)( 83,144)( 84,143);;
s2 := (169,170);;
s3 := (171,172);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, 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*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*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*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*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*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*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*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(172)!(  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)
( 12, 18)( 13, 17)( 14, 16)( 23, 28)( 24, 27)( 25, 26)( 29, 36)( 30, 42)
( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 43, 64)( 44, 70)( 45, 69)
( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 78)( 51, 84)( 52, 83)( 53, 82)
( 54, 81)( 55, 80)( 56, 79)( 57, 71)( 58, 77)( 59, 76)( 60, 75)( 61, 74)
( 62, 73)( 63, 72)( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)
( 91,128)( 92,141)( 93,147)( 94,146)( 95,145)( 96,144)( 97,143)( 98,142)
( 99,134)(100,140)(101,139)(102,138)(103,137)(104,136)(105,135)(106,148)
(107,154)(108,153)(109,152)(110,151)(111,150)(112,149)(113,162)(114,168)
(115,167)(116,166)(117,165)(118,164)(119,163)(120,155)(121,161)(122,160)
(123,159)(124,158)(125,157)(126,156);
s1 := Sym(172)!(  1, 93)(  2, 92)(  3, 98)(  4, 97)(  5, 96)(  6, 95)(  7, 94)
(  8, 86)(  9, 85)( 10, 91)( 11, 90)( 12, 89)( 13, 88)( 14, 87)( 15,100)
( 16, 99)( 17,105)( 18,104)( 19,103)( 20,102)( 21,101)( 22,114)( 23,113)
( 24,119)( 25,118)( 26,117)( 27,116)( 28,115)( 29,107)( 30,106)( 31,112)
( 32,111)( 33,110)( 34,109)( 35,108)( 36,121)( 37,120)( 38,126)( 39,125)
( 40,124)( 41,123)( 42,122)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,149)( 51,148)( 52,154)( 53,153)( 54,152)( 55,151)
( 56,150)( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)
( 64,135)( 65,134)( 66,140)( 67,139)( 68,138)( 69,137)( 70,136)( 71,128)
( 72,127)( 73,133)( 74,132)( 75,131)( 76,130)( 77,129)( 78,142)( 79,141)
( 80,147)( 81,146)( 82,145)( 83,144)( 84,143);
s2 := Sym(172)!(169,170);
s3 := Sym(172)!(171,172);
poly := sub<Sym(172)|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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, 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*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*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*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*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*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*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*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 >; 
 

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