Polytope of Type {3,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,14}*672
Also Known As : {{3,6}₄,{6,14|2}}. if this polytope has another name.
Group : SmallGroup(672,1260)
Rank : 4
Schlafli Type : {3,6,14}
Number of vertices, edges, etc : 4, 12, 56, 14
Order of s₀s₁s₂s₃ : 28
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,14,2} of size 1344
Vertex Figure Of :
   {2,3,6,14} of size 1344
Quotients (Maximal Quotients in Boldface) :
   7-fold quotients : {3,6,2}*96
   14-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,28}*1344, {3,12,14}*1344, {6,6,14}*1344
Permutation Representation (GAP) :

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

References : None.
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