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Polytope of Type {2,14,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,14,4,3}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 5
Schlafli Type : {2,14,4,3}
Number of vertices, edges, etc : 2, 14, 56, 12, 6
Order of s0s1s2s3s4 : 42
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   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) :
   4-fold quotients : {2,14,2,3}*336
   7-fold quotients : {2,2,4,3}*192
   8-fold quotients : {2,7,2,3}*168
   14-fold quotients : {2,2,4,3}*96
   28-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  7, 27)(  8, 28)(  9, 29)( 10, 30)( 11, 23)( 12, 24)( 13, 25)( 14, 26)
( 15, 19)( 16, 20)( 17, 21)( 18, 22)( 35, 55)( 36, 56)( 37, 57)( 38, 58)
( 39, 51)( 40, 52)( 41, 53)( 42, 54)( 43, 47)( 44, 48)( 45, 49)( 46, 50)
( 63, 83)( 64, 84)( 65, 85)( 66, 86)( 67, 79)( 68, 80)( 69, 81)( 70, 82)
( 71, 75)( 72, 76)( 73, 77)( 74, 78)( 91,111)( 92,112)( 93,113)( 94,114)
( 95,107)( 96,108)( 97,109)( 98,110)( 99,103)(100,104)(101,105)(102,106)
(119,139)(120,140)(121,141)(122,142)(123,135)(124,136)(125,137)(126,138)
(127,131)(128,132)(129,133)(130,134)(147,167)(148,168)(149,169)(150,170)
(151,163)(152,164)(153,165)(154,166)(155,159)(156,160)(157,161)(158,162);;
s2 := (  3, 93)(  4, 94)(  5, 91)(  6, 92)(  7, 89)(  8, 90)(  9, 87)( 10, 88)
( 11,113)( 12,114)( 13,111)( 14,112)( 15,109)( 16,110)( 17,107)( 18,108)
( 19,105)( 20,106)( 21,103)( 22,104)( 23,101)( 24,102)( 25, 99)( 26,100)
( 27, 97)( 28, 98)( 29, 95)( 30, 96)( 31,121)( 32,122)( 33,119)( 34,120)
( 35,117)( 36,118)( 37,115)( 38,116)( 39,141)( 40,142)( 41,139)( 42,140)
( 43,137)( 44,138)( 45,135)( 46,136)( 47,133)( 48,134)( 49,131)( 50,132)
( 51,129)( 52,130)( 53,127)( 54,128)( 55,125)( 56,126)( 57,123)( 58,124)
( 59,149)( 60,150)( 61,147)( 62,148)( 63,145)( 64,146)( 65,143)( 66,144)
( 67,169)( 68,170)( 69,167)( 70,168)( 71,165)( 72,166)( 73,163)( 74,164)
( 75,161)( 76,162)( 77,159)( 78,160)( 79,157)( 80,158)( 81,155)( 82,156)
( 83,153)( 84,154)( 85,151)( 86,152);;
s3 := (  4,  5)(  8,  9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)( 31, 59)
( 32, 61)( 33, 60)( 34, 62)( 35, 63)( 36, 65)( 37, 64)( 38, 66)( 39, 67)
( 40, 69)( 41, 68)( 42, 70)( 43, 71)( 44, 73)( 45, 72)( 46, 74)( 47, 75)
( 48, 77)( 49, 76)( 50, 78)( 51, 79)( 52, 81)( 53, 80)( 54, 82)( 55, 83)
( 56, 85)( 57, 84)( 58, 86)( 88, 89)( 92, 93)( 96, 97)(100,101)(104,105)
(108,109)(112,113)(115,143)(116,145)(117,144)(118,146)(119,147)(120,149)
(121,148)(122,150)(123,151)(124,153)(125,152)(126,154)(127,155)(128,157)
(129,156)(130,158)(131,159)(132,161)(133,160)(134,162)(135,163)(136,165)
(137,164)(138,166)(139,167)(140,169)(141,168)(142,170);;
s4 := (  3, 59)(  4, 62)(  5, 61)(  6, 60)(  7, 63)(  8, 66)(  9, 65)( 10, 64)
( 11, 67)( 12, 70)( 13, 69)( 14, 68)( 15, 71)( 16, 74)( 17, 73)( 18, 72)
( 19, 75)( 20, 78)( 21, 77)( 22, 76)( 23, 79)( 24, 82)( 25, 81)( 26, 80)
( 27, 83)( 28, 86)( 29, 85)( 30, 84)( 32, 34)( 36, 38)( 40, 42)( 44, 46)
( 48, 50)( 52, 54)( 56, 58)( 87,143)( 88,146)( 89,145)( 90,144)( 91,147)
( 92,150)( 93,149)( 94,148)( 95,151)( 96,154)( 97,153)( 98,152)( 99,155)
(100,158)(101,157)(102,156)(103,159)(104,162)(105,161)(106,160)(107,163)
(108,166)(109,165)(110,164)(111,167)(112,170)(113,169)(114,168)(116,118)
(120,122)(124,126)(128,130)(132,134)(136,138)(140,142);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
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(170)!(1,2);
s1 := Sym(170)!(  7, 27)(  8, 28)(  9, 29)( 10, 30)( 11, 23)( 12, 24)( 13, 25)
( 14, 26)( 15, 19)( 16, 20)( 17, 21)( 18, 22)( 35, 55)( 36, 56)( 37, 57)
( 38, 58)( 39, 51)( 40, 52)( 41, 53)( 42, 54)( 43, 47)( 44, 48)( 45, 49)
( 46, 50)( 63, 83)( 64, 84)( 65, 85)( 66, 86)( 67, 79)( 68, 80)( 69, 81)
( 70, 82)( 71, 75)( 72, 76)( 73, 77)( 74, 78)( 91,111)( 92,112)( 93,113)
( 94,114)( 95,107)( 96,108)( 97,109)( 98,110)( 99,103)(100,104)(101,105)
(102,106)(119,139)(120,140)(121,141)(122,142)(123,135)(124,136)(125,137)
(126,138)(127,131)(128,132)(129,133)(130,134)(147,167)(148,168)(149,169)
(150,170)(151,163)(152,164)(153,165)(154,166)(155,159)(156,160)(157,161)
(158,162);
s2 := Sym(170)!(  3, 93)(  4, 94)(  5, 91)(  6, 92)(  7, 89)(  8, 90)(  9, 87)
( 10, 88)( 11,113)( 12,114)( 13,111)( 14,112)( 15,109)( 16,110)( 17,107)
( 18,108)( 19,105)( 20,106)( 21,103)( 22,104)( 23,101)( 24,102)( 25, 99)
( 26,100)( 27, 97)( 28, 98)( 29, 95)( 30, 96)( 31,121)( 32,122)( 33,119)
( 34,120)( 35,117)( 36,118)( 37,115)( 38,116)( 39,141)( 40,142)( 41,139)
( 42,140)( 43,137)( 44,138)( 45,135)( 46,136)( 47,133)( 48,134)( 49,131)
( 50,132)( 51,129)( 52,130)( 53,127)( 54,128)( 55,125)( 56,126)( 57,123)
( 58,124)( 59,149)( 60,150)( 61,147)( 62,148)( 63,145)( 64,146)( 65,143)
( 66,144)( 67,169)( 68,170)( 69,167)( 70,168)( 71,165)( 72,166)( 73,163)
( 74,164)( 75,161)( 76,162)( 77,159)( 78,160)( 79,157)( 80,158)( 81,155)
( 82,156)( 83,153)( 84,154)( 85,151)( 86,152);
s3 := Sym(170)!(  4,  5)(  8,  9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)
( 31, 59)( 32, 61)( 33, 60)( 34, 62)( 35, 63)( 36, 65)( 37, 64)( 38, 66)
( 39, 67)( 40, 69)( 41, 68)( 42, 70)( 43, 71)( 44, 73)( 45, 72)( 46, 74)
( 47, 75)( 48, 77)( 49, 76)( 50, 78)( 51, 79)( 52, 81)( 53, 80)( 54, 82)
( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 88, 89)( 92, 93)( 96, 97)(100,101)
(104,105)(108,109)(112,113)(115,143)(116,145)(117,144)(118,146)(119,147)
(120,149)(121,148)(122,150)(123,151)(124,153)(125,152)(126,154)(127,155)
(128,157)(129,156)(130,158)(131,159)(132,161)(133,160)(134,162)(135,163)
(136,165)(137,164)(138,166)(139,167)(140,169)(141,168)(142,170);
s4 := Sym(170)!(  3, 59)(  4, 62)(  5, 61)(  6, 60)(  7, 63)(  8, 66)(  9, 65)
( 10, 64)( 11, 67)( 12, 70)( 13, 69)( 14, 68)( 15, 71)( 16, 74)( 17, 73)
( 18, 72)( 19, 75)( 20, 78)( 21, 77)( 22, 76)( 23, 79)( 24, 82)( 25, 81)
( 26, 80)( 27, 83)( 28, 86)( 29, 85)( 30, 84)( 32, 34)( 36, 38)( 40, 42)
( 44, 46)( 48, 50)( 52, 54)( 56, 58)( 87,143)( 88,146)( 89,145)( 90,144)
( 91,147)( 92,150)( 93,149)( 94,148)( 95,151)( 96,154)( 97,153)( 98,152)
( 99,155)(100,158)(101,157)(102,156)(103,159)(104,162)(105,161)(106,160)
(107,163)(108,166)(109,165)(110,164)(111,167)(112,170)(113,169)(114,168)
(116,118)(120,122)(124,126)(128,130)(132,134)(136,138)(140,142);
poly := sub<Sym(170)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, 
s1*s2*s3*s2*s1*s2*s3*s2, 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 >; 
 

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