Polytope of Type {6,14,8}

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

References : None.
to this polytope