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Polytope of Type {14,6,8}

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