Polytope of Type {8,6,14}

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

References : None.
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