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Polytope of Type {6,8,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,14}*1344
Also Known As : {{6,8|2},{8,14|2}}. if this polytope has another name.
Group : SmallGroup(1344,8561)
Rank : 4
Schlafli Type : {6,8,14}
Number of vertices, edges, etc : 6, 24, 56, 14
Order of s0s1s2s3 : 168
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {6,4,14}*672
   3-fold quotients : {2,8,14}*448
   4-fold quotients : {6,2,14}*336
   6-fold quotients : {2,4,14}*224
   7-fold quotients : {6,8,2}*192
   8-fold quotients : {3,2,14}*168, {6,2,7}*168
   12-fold quotients : {2,2,14}*112
   14-fold quotients : {6,4,2}*96a
   16-fold quotients : {3,2,7}*84
   21-fold quotients : {2,8,2}*64
   24-fold quotients : {2,2,7}*56
   28-fold quotients : {6,2,2}*48
   42-fold quotients : {2,4,2}*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 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)( 51, 58)
( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)( 95,102)
( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)
(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)(139,146)
(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168);;
s1 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 22, 29)
( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 71)( 44, 72)
( 45, 73)( 46, 74)( 47, 75)( 48, 76)( 49, 77)( 50, 64)( 51, 65)( 52, 66)
( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 57, 78)( 58, 79)( 59, 80)( 60, 81)
( 61, 82)( 62, 83)( 63, 84)( 85,134)( 86,135)( 87,136)( 88,137)( 89,138)
( 90,139)( 91,140)( 92,127)( 93,128)( 94,129)( 95,130)( 96,131)( 97,132)
( 98,133)( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)
(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,148)
(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,162)(121,163)
(122,164)(123,165)(124,166)(125,167)(126,168);;
s2 := (  1, 85)(  2, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)(  8, 92)
(  9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 93)( 15, 99)( 16,105)
( 17,104)( 18,103)( 19,102)( 20,101)( 21,100)( 22,106)( 23,112)( 24,111)
( 25,110)( 26,109)( 27,108)( 28,107)( 29,113)( 30,119)( 31,118)( 32,117)
( 33,116)( 34,115)( 35,114)( 36,120)( 37,126)( 38,125)( 39,124)( 40,123)
( 41,122)( 42,121)( 43,148)( 44,154)( 45,153)( 46,152)( 47,151)( 48,150)
( 49,149)( 50,155)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)( 56,156)
( 57,162)( 58,168)( 59,167)( 60,166)( 61,165)( 62,164)( 63,163)( 64,127)
( 65,133)( 66,132)( 67,131)( 68,130)( 69,129)( 70,128)( 71,134)( 72,140)
( 73,139)( 74,138)( 75,137)( 76,136)( 77,135)( 78,141)( 79,147)( 80,146)
( 81,145)( 82,144)( 83,143)( 84,142);;
s3 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)
( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)(109,111)
(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)(129,133)
(130,132)(134,135)(136,140)(137,139)(141,142)(143,147)(144,146)(148,149)
(150,154)(151,153)(155,156)(157,161)(158,160)(162,163)(164,168)(165,167);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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)!(  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)
( 29, 36)( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 50, 57)
( 51, 58)( 52, 59)( 53, 60)( 54, 61)( 55, 62)( 56, 63)( 71, 78)( 72, 79)
( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 92, 99)( 93,100)( 94,101)
( 95,102)( 96,103)( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)
(117,124)(118,125)(119,126)(134,141)(135,142)(136,143)(137,144)(138,145)
(139,146)(140,147)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)
(161,168);
s1 := Sym(168)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 22, 29)( 23, 30)( 24, 31)( 25, 32)( 26, 33)( 27, 34)( 28, 35)( 43, 71)
( 44, 72)( 45, 73)( 46, 74)( 47, 75)( 48, 76)( 49, 77)( 50, 64)( 51, 65)
( 52, 66)( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 57, 78)( 58, 79)( 59, 80)
( 60, 81)( 61, 82)( 62, 83)( 63, 84)( 85,134)( 86,135)( 87,136)( 88,137)
( 89,138)( 90,139)( 91,140)( 92,127)( 93,128)( 94,129)( 95,130)( 96,131)
( 97,132)( 98,133)( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)
(105,147)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)
(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,162)
(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);
s2 := Sym(168)!(  1, 85)(  2, 91)(  3, 90)(  4, 89)(  5, 88)(  6, 87)(  7, 86)
(  8, 92)(  9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 93)( 15, 99)
( 16,105)( 17,104)( 18,103)( 19,102)( 20,101)( 21,100)( 22,106)( 23,112)
( 24,111)( 25,110)( 26,109)( 27,108)( 28,107)( 29,113)( 30,119)( 31,118)
( 32,117)( 33,116)( 34,115)( 35,114)( 36,120)( 37,126)( 38,125)( 39,124)
( 40,123)( 41,122)( 42,121)( 43,148)( 44,154)( 45,153)( 46,152)( 47,151)
( 48,150)( 49,149)( 50,155)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)
( 56,156)( 57,162)( 58,168)( 59,167)( 60,166)( 61,165)( 62,164)( 63,163)
( 64,127)( 65,133)( 66,132)( 67,131)( 68,130)( 69,129)( 70,128)( 71,134)
( 72,140)( 73,139)( 74,138)( 75,137)( 76,136)( 77,135)( 78,141)( 79,147)
( 80,146)( 81,145)( 82,144)( 83,143)( 84,142);
s3 := Sym(168)!(  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)
( 53, 55)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)
( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)
(109,111)(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)
(129,133)(130,132)(134,135)(136,140)(137,139)(141,142)(143,147)(144,146)
(148,149)(150,154)(151,153)(155,156)(157,161)(158,160)(162,163)(164,168)
(165,167);
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*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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.
to this polytope