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Polytope of Type {14,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,4,3}*672
Also Known As : {{14,4|2},{4,3}}. if this polytope has another name.
Group : SmallGroup(672,1260)
Rank : 4
Schlafli Type : {14,4,3}
Number of vertices, edges, etc : 14, 56, 12, 6
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,4,3,2} of size 1344
Vertex Figure Of :
   {2,14,4,3} of size 1344
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {14,2,3}*168
   7-fold quotients : {2,4,3}*96
   8-fold quotients : {7,2,3}*84
   14-fold quotients : {2,4,3}*48
   28-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {28,4,3}*1344, {14,8,3}*1344, {14,4,6}*1344
Permutation Representation (GAP) :
s0 := (  5, 25)(  6, 26)(  7, 27)(  8, 28)(  9, 21)( 10, 22)( 11, 23)( 12, 24)
( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 33, 53)( 34, 54)( 35, 55)( 36, 56)
( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 77)( 66, 78)( 67, 79)( 68, 80)
( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 89,109)( 90,110)( 91,111)( 92,112)
( 93,105)( 94,106)( 95,107)( 96,108)( 97,101)( 98,102)( 99,103)(100,104)
(117,137)(118,138)(119,139)(120,140)(121,133)(122,134)(123,135)(124,136)
(125,129)(126,130)(127,131)(128,132)(145,165)(146,166)(147,167)(148,168)
(149,161)(150,162)(151,163)(152,164)(153,157)(154,158)(155,159)(156,160);;
s1 := (  1, 91)(  2, 92)(  3, 89)(  4, 90)(  5, 87)(  6, 88)(  7, 85)(  8, 86)
(  9,111)( 10,112)( 11,109)( 12,110)( 13,107)( 14,108)( 15,105)( 16,106)
( 17,103)( 18,104)( 19,101)( 20,102)( 21, 99)( 22,100)( 23, 97)( 24, 98)
( 25, 95)( 26, 96)( 27, 93)( 28, 94)( 29,119)( 30,120)( 31,117)( 32,118)
( 33,115)( 34,116)( 35,113)( 36,114)( 37,139)( 38,140)( 39,137)( 40,138)
( 41,135)( 42,136)( 43,133)( 44,134)( 45,131)( 46,132)( 47,129)( 48,130)
( 49,127)( 50,128)( 51,125)( 52,126)( 53,123)( 54,124)( 55,121)( 56,122)
( 57,147)( 58,148)( 59,145)( 60,146)( 61,143)( 62,144)( 63,141)( 64,142)
( 65,167)( 66,168)( 67,165)( 68,166)( 69,163)( 70,164)( 71,161)( 72,162)
( 73,159)( 74,160)( 75,157)( 76,158)( 77,155)( 78,156)( 79,153)( 80,154)
( 81,151)( 82,152)( 83,149)( 84,150);;
s2 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)( 29, 57)
( 30, 59)( 31, 58)( 32, 60)( 33, 61)( 34, 63)( 35, 62)( 36, 64)( 37, 65)
( 38, 67)( 39, 66)( 40, 68)( 41, 69)( 42, 71)( 43, 70)( 44, 72)( 45, 73)
( 46, 75)( 47, 74)( 48, 76)( 49, 77)( 50, 79)( 51, 78)( 52, 80)( 53, 81)
( 54, 83)( 55, 82)( 56, 84)( 86, 87)( 90, 91)( 94, 95)( 98, 99)(102,103)
(106,107)(110,111)(113,141)(114,143)(115,142)(116,144)(117,145)(118,147)
(119,146)(120,148)(121,149)(122,151)(123,150)(124,152)(125,153)(126,155)
(127,154)(128,156)(129,157)(130,159)(131,158)(132,160)(133,161)(134,163)
(135,162)(136,164)(137,165)(138,167)(139,166)(140,168);;
s3 := (  1, 57)(  2, 60)(  3, 59)(  4, 58)(  5, 61)(  6, 64)(  7, 63)(  8, 62)
(  9, 65)( 10, 68)( 11, 67)( 12, 66)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 77)( 22, 80)( 23, 79)( 24, 78)
( 25, 81)( 26, 84)( 27, 83)( 28, 82)( 30, 32)( 34, 36)( 38, 40)( 42, 44)
( 46, 48)( 50, 52)( 54, 56)( 85,141)( 86,144)( 87,143)( 88,142)( 89,145)
( 90,148)( 91,147)( 92,146)( 93,149)( 94,152)( 95,151)( 96,150)( 97,153)
( 98,156)( 99,155)(100,154)(101,157)(102,160)(103,159)(104,158)(105,161)
(106,164)(107,163)(108,162)(109,165)(110,168)(111,167)(112,166)(114,116)
(118,120)(122,124)(126,128)(130,132)(134,136)(138,140);;
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, s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(168)!(  5, 25)(  6, 26)(  7, 27)(  8, 28)(  9, 21)( 10, 22)( 11, 23)
( 12, 24)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 33, 53)( 34, 54)( 35, 55)
( 36, 56)( 37, 49)( 38, 50)( 39, 51)( 40, 52)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 61, 81)( 62, 82)( 63, 83)( 64, 84)( 65, 77)( 66, 78)( 67, 79)
( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 89,109)( 90,110)( 91,111)
( 92,112)( 93,105)( 94,106)( 95,107)( 96,108)( 97,101)( 98,102)( 99,103)
(100,104)(117,137)(118,138)(119,139)(120,140)(121,133)(122,134)(123,135)
(124,136)(125,129)(126,130)(127,131)(128,132)(145,165)(146,166)(147,167)
(148,168)(149,161)(150,162)(151,163)(152,164)(153,157)(154,158)(155,159)
(156,160);
s1 := Sym(168)!(  1, 91)(  2, 92)(  3, 89)(  4, 90)(  5, 87)(  6, 88)(  7, 85)
(  8, 86)(  9,111)( 10,112)( 11,109)( 12,110)( 13,107)( 14,108)( 15,105)
( 16,106)( 17,103)( 18,104)( 19,101)( 20,102)( 21, 99)( 22,100)( 23, 97)
( 24, 98)( 25, 95)( 26, 96)( 27, 93)( 28, 94)( 29,119)( 30,120)( 31,117)
( 32,118)( 33,115)( 34,116)( 35,113)( 36,114)( 37,139)( 38,140)( 39,137)
( 40,138)( 41,135)( 42,136)( 43,133)( 44,134)( 45,131)( 46,132)( 47,129)
( 48,130)( 49,127)( 50,128)( 51,125)( 52,126)( 53,123)( 54,124)( 55,121)
( 56,122)( 57,147)( 58,148)( 59,145)( 60,146)( 61,143)( 62,144)( 63,141)
( 64,142)( 65,167)( 66,168)( 67,165)( 68,166)( 69,163)( 70,164)( 71,161)
( 72,162)( 73,159)( 74,160)( 75,157)( 76,158)( 77,155)( 78,156)( 79,153)
( 80,154)( 81,151)( 82,152)( 83,149)( 84,150);
s2 := Sym(168)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 22, 23)( 26, 27)
( 29, 57)( 30, 59)( 31, 58)( 32, 60)( 33, 61)( 34, 63)( 35, 62)( 36, 64)
( 37, 65)( 38, 67)( 39, 66)( 40, 68)( 41, 69)( 42, 71)( 43, 70)( 44, 72)
( 45, 73)( 46, 75)( 47, 74)( 48, 76)( 49, 77)( 50, 79)( 51, 78)( 52, 80)
( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 86, 87)( 90, 91)( 94, 95)( 98, 99)
(102,103)(106,107)(110,111)(113,141)(114,143)(115,142)(116,144)(117,145)
(118,147)(119,146)(120,148)(121,149)(122,151)(123,150)(124,152)(125,153)
(126,155)(127,154)(128,156)(129,157)(130,159)(131,158)(132,160)(133,161)
(134,163)(135,162)(136,164)(137,165)(138,167)(139,166)(140,168);
s3 := Sym(168)!(  1, 57)(  2, 60)(  3, 59)(  4, 58)(  5, 61)(  6, 64)(  7, 63)
(  8, 62)(  9, 65)( 10, 68)( 11, 67)( 12, 66)( 13, 69)( 14, 72)( 15, 71)
( 16, 70)( 17, 73)( 18, 76)( 19, 75)( 20, 74)( 21, 77)( 22, 80)( 23, 79)
( 24, 78)( 25, 81)( 26, 84)( 27, 83)( 28, 82)( 30, 32)( 34, 36)( 38, 40)
( 42, 44)( 46, 48)( 50, 52)( 54, 56)( 85,141)( 86,144)( 87,143)( 88,142)
( 89,145)( 90,148)( 91,147)( 92,146)( 93,149)( 94,152)( 95,151)( 96,150)
( 97,153)( 98,156)( 99,155)(100,154)(101,157)(102,160)(103,159)(104,158)
(105,161)(106,164)(107,163)(108,162)(109,165)(110,168)(111,167)(112,166)
(114,116)(118,120)(122,124)(126,128)(130,132)(134,136)(138,140);
poly := sub<Sym(168)|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, 
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*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 >; 
 
References : None.
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