Polytope of Type {8,14,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,14,6}*1344
Also Known As : {{8,14|2},{14,6|2}}. if this polytope has another name.
Group : SmallGroup(1344,8561)
Rank : 4
Schlafli Type : {8,14,6}
Number of vertices, edges, etc : 8, 56, 42, 6
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 : {4,14,6}*672
   3-fold quotients : {8,14,2}*448
   4-fold quotients : {2,14,6}*336
   6-fold quotients : {4,14,2}*224
   7-fold quotients : {8,2,6}*192
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {8,2,3}*96, {4,2,6}*96
   21-fold quotients : {8,2,2}*64
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {4,2,3}*48, {2,2,6}*48
   42-fold quotients : {4,2,2}*32
   56-fold quotients : {2,2,3}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 71)
( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)( 58, 79)
( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)( 85,127)( 86,128)( 87,129)
( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)( 95,137)
( 96,138)( 97,139)( 98,140)( 99,141)(100,142)(101,143)(102,144)(103,145)
(104,146)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)
(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)
(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);;
s1 := (  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);;
s2 := (  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);;
s3 := (  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, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)( 88, 95)
( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)(110,117)
(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)(132,139)
(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161);;
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, 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, 
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)!( 43, 64)( 44, 65)( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)
( 50, 71)( 51, 72)( 52, 73)( 53, 74)( 54, 75)( 55, 76)( 56, 77)( 57, 78)
( 58, 79)( 59, 80)( 60, 81)( 61, 82)( 62, 83)( 63, 84)( 85,127)( 86,128)
( 87,129)( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)
( 95,137)( 96,138)( 97,139)( 98,140)( 99,141)(100,142)(101,143)(102,144)
(103,145)(104,146)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)
(111,153)(112,154)(113,155)(114,156)(115,157)(116,158)(117,159)(118,160)
(119,161)(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168);
s1 := 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);
s2 := 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);
s3 := 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, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85, 92)( 86, 93)( 87, 94)
( 88, 95)( 89, 96)( 90, 97)( 91, 98)(106,113)(107,114)(108,115)(109,116)
(110,117)(111,118)(112,119)(127,134)(128,135)(129,136)(130,137)(131,138)
(132,139)(133,140)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)
(154,161);
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, 
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, 
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