Polytope of Type {14,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,3}*672
Also Known As : {{14,6|2},{6,3}₄}. if this polytope has another name.
Group : SmallGroup(672,1260)
Rank : 4
Schlafli Type : {14,6,3}
Number of vertices, edges, etc : 14, 56, 12, 4
Order of s₀s₁s₂s₃ : 28
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,6,3,2} of size 1344
Vertex Figure Of :
   {2,14,6,3} of size 1344
Quotients (Maximal Quotients in Boldface) :
   7-fold quotients : {2,6,3}*96
   14-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {28,6,3}*1344, {14,12,3}*1344, {14,6,6}*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,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 25)( 10, 27)( 11, 26)( 12, 28)
( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 29, 61)( 30, 63)( 31, 62)
( 32, 64)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 81)( 38, 83)( 39, 82)
( 40, 84)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 73)( 46, 75)( 47, 74)
( 48, 76)( 49, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)
( 56, 68)( 85, 89)( 86, 91)( 87, 90)( 88, 92)( 93,109)( 94,111)( 95,110)
( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(113,145)(114,147)
(115,146)(116,148)(117,141)(118,143)(119,142)(120,144)(121,165)(122,167)
(123,166)(124,168)(125,161)(126,163)(127,162)(128,164)(129,157)(130,159)
(131,158)(132,160)(133,153)(134,155)(135,154)(136,156)(137,149)(138,151)
(139,150)(140,152);;
s2 := (  1,113)(  2,114)(  3,116)(  4,115)(  5,117)(  6,118)(  7,120)(  8,119)
(  9,121)( 10,122)( 11,124)( 12,123)( 13,125)( 14,126)( 15,128)( 16,127)
( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)( 24,135)
( 25,137)( 26,138)( 27,140)( 28,139)( 29, 85)( 30, 86)( 31, 88)( 32, 87)
( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 93)( 38, 94)( 39, 96)( 40, 95)
( 41, 97)( 42, 98)( 43,100)( 44, 99)( 45,101)( 46,102)( 47,104)( 48,103)
( 49,105)( 50,106)( 51,108)( 52,107)( 53,109)( 54,110)( 55,112)( 56,111)
( 57,141)( 58,142)( 59,144)( 60,143)( 61,145)( 62,146)( 63,148)( 64,147)
( 65,149)( 66,150)( 67,152)( 68,151)( 69,153)( 70,154)( 71,156)( 72,155)
( 73,157)( 74,158)( 75,160)( 76,159)( 77,161)( 78,162)( 79,164)( 80,163)
( 81,165)( 82,166)( 83,168)( 84,167);;
s3 := (  1, 88)(  2, 86)(  3, 87)(  4, 85)(  5, 92)(  6, 90)(  7, 91)(  8, 89)
(  9, 96)( 10, 94)( 11, 95)( 12, 93)( 13,100)( 14, 98)( 15, 99)( 16, 97)
( 17,104)( 18,102)( 19,103)( 20,101)( 21,108)( 22,106)( 23,107)( 24,105)
( 25,112)( 26,110)( 27,111)( 28,109)( 29,144)( 30,142)( 31,143)( 32,141)
( 33,148)( 34,146)( 35,147)( 36,145)( 37,152)( 38,150)( 39,151)( 40,149)
( 41,156)( 42,154)( 43,155)( 44,153)( 45,160)( 46,158)( 47,159)( 48,157)
( 49,164)( 50,162)( 51,163)( 52,161)( 53,168)( 54,166)( 55,167)( 56,165)
( 57,116)( 58,114)( 59,115)( 60,113)( 61,120)( 62,118)( 63,119)( 64,117)
( 65,124)( 66,122)( 67,123)( 68,121)( 69,128)( 70,126)( 71,127)( 72,125)
( 73,132)( 74,130)( 75,131)( 76,129)( 77,136)( 78,134)( 79,135)( 80,133)
( 81,140)( 82,138)( 83,139)( 84,137);;
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*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*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,  5)(  2,  7)(  3,  6)(  4,  8)(  9, 25)( 10, 27)( 11, 26)
( 12, 28)( 13, 21)( 14, 23)( 15, 22)( 16, 24)( 18, 19)( 29, 61)( 30, 63)
( 31, 62)( 32, 64)( 33, 57)( 34, 59)( 35, 58)( 36, 60)( 37, 81)( 38, 83)
( 39, 82)( 40, 84)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 73)( 46, 75)
( 47, 74)( 48, 76)( 49, 69)( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 85, 89)( 86, 91)( 87, 90)( 88, 92)( 93,109)( 94,111)
( 95,110)( 96,112)( 97,105)( 98,107)( 99,106)(100,108)(102,103)(113,145)
(114,147)(115,146)(116,148)(117,141)(118,143)(119,142)(120,144)(121,165)
(122,167)(123,166)(124,168)(125,161)(126,163)(127,162)(128,164)(129,157)
(130,159)(131,158)(132,160)(133,153)(134,155)(135,154)(136,156)(137,149)
(138,151)(139,150)(140,152);
s2 := Sym(168)!(  1,113)(  2,114)(  3,116)(  4,115)(  5,117)(  6,118)(  7,120)
(  8,119)(  9,121)( 10,122)( 11,124)( 12,123)( 13,125)( 14,126)( 15,128)
( 16,127)( 17,129)( 18,130)( 19,132)( 20,131)( 21,133)( 22,134)( 23,136)
( 24,135)( 25,137)( 26,138)( 27,140)( 28,139)( 29, 85)( 30, 86)( 31, 88)
( 32, 87)( 33, 89)( 34, 90)( 35, 92)( 36, 91)( 37, 93)( 38, 94)( 39, 96)
( 40, 95)( 41, 97)( 42, 98)( 43,100)( 44, 99)( 45,101)( 46,102)( 47,104)
( 48,103)( 49,105)( 50,106)( 51,108)( 52,107)( 53,109)( 54,110)( 55,112)
( 56,111)( 57,141)( 58,142)( 59,144)( 60,143)( 61,145)( 62,146)( 63,148)
( 64,147)( 65,149)( 66,150)( 67,152)( 68,151)( 69,153)( 70,154)( 71,156)
( 72,155)( 73,157)( 74,158)( 75,160)( 76,159)( 77,161)( 78,162)( 79,164)
( 80,163)( 81,165)( 82,166)( 83,168)( 84,167);
s3 := Sym(168)!(  1, 88)(  2, 86)(  3, 87)(  4, 85)(  5, 92)(  6, 90)(  7, 91)
(  8, 89)(  9, 96)( 10, 94)( 11, 95)( 12, 93)( 13,100)( 14, 98)( 15, 99)
( 16, 97)( 17,104)( 18,102)( 19,103)( 20,101)( 21,108)( 22,106)( 23,107)
( 24,105)( 25,112)( 26,110)( 27,111)( 28,109)( 29,144)( 30,142)( 31,143)
( 32,141)( 33,148)( 34,146)( 35,147)( 36,145)( 37,152)( 38,150)( 39,151)
( 40,149)( 41,156)( 42,154)( 43,155)( 44,153)( 45,160)( 46,158)( 47,159)
( 48,157)( 49,164)( 50,162)( 51,163)( 52,161)( 53,168)( 54,166)( 55,167)
( 56,165)( 57,116)( 58,114)( 59,115)( 60,113)( 61,120)( 62,118)( 63,119)
( 64,117)( 65,124)( 66,122)( 67,123)( 68,121)( 69,128)( 70,126)( 71,127)
( 72,125)( 73,132)( 74,130)( 75,131)( 76,129)( 77,136)( 78,134)( 79,135)
( 80,133)( 81,140)( 82,138)( 83,139)( 84,137);
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*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2, 
s1*s2*s1*s2*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.
to this polytope