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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,4,14}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 5
Schlafli Type : {2,3,4,14}
Number of vertices, edges, etc : 2, 6, 12, 56, 14
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 : {2,3,2,14}*336
   7-fold quotients : {2,3,4,2}*192
   8-fold quotients : {2,3,2,7}*168
   14-fold quotients : {2,3,4,2}*96
   28-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  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);;
s2 := (  3, 31)(  4, 32)(  5, 34)(  6, 33)(  7, 35)(  8, 36)(  9, 38)( 10, 37)
( 11, 39)( 12, 40)( 13, 42)( 14, 41)( 15, 43)( 16, 44)( 17, 46)( 18, 45)
( 19, 47)( 20, 48)( 21, 50)( 22, 49)( 23, 51)( 24, 52)( 25, 54)( 26, 53)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 61, 62)( 65, 66)( 69, 70)( 73, 74)
( 77, 78)( 81, 82)( 85, 86)( 87,115)( 88,116)( 89,118)( 90,117)( 91,119)
( 92,120)( 93,122)( 94,121)( 95,123)( 96,124)( 97,126)( 98,125)( 99,127)
(100,128)(101,130)(102,129)(103,131)(104,132)(105,134)(106,133)(107,135)
(108,136)(109,138)(110,137)(111,139)(112,140)(113,142)(114,141)(145,146)
(149,150)(153,154)(157,158)(161,162)(165,166)(169,170);;
s3 := (  3,  6)(  4,  5)(  7, 30)(  8, 29)(  9, 28)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 31, 34)( 32, 33)
( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)
( 43, 50)( 44, 49)( 45, 48)( 46, 47)( 59, 62)( 60, 61)( 63, 86)( 64, 85)
( 65, 84)( 66, 83)( 67, 82)( 68, 81)( 69, 80)( 70, 79)( 71, 78)( 72, 77)
( 73, 76)( 74, 75)( 87, 90)( 88, 89)( 91,114)( 92,113)( 93,112)( 94,111)
( 95,110)( 96,109)( 97,108)( 98,107)( 99,106)(100,105)(101,104)(102,103)
(115,118)(116,117)(119,142)(120,141)(121,140)(122,139)(123,138)(124,137)
(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(143,146)(144,145)
(147,170)(148,169)(149,168)(150,167)(151,166)(152,165)(153,164)(154,163)
(155,162)(156,161)(157,160)(158,159);;
s4 := (  3, 91)(  4, 92)(  5, 93)(  6, 94)(  7, 87)(  8, 88)(  9, 89)( 10, 90)
( 11,111)( 12,112)( 13,113)( 14,114)( 15,107)( 16,108)( 17,109)( 18,110)
( 19,103)( 20,104)( 21,105)( 22,106)( 23, 99)( 24,100)( 25,101)( 26,102)
( 27, 95)( 28, 96)( 29, 97)( 30, 98)( 31,119)( 32,120)( 33,121)( 34,122)
( 35,115)( 36,116)( 37,117)( 38,118)( 39,139)( 40,140)( 41,141)( 42,142)
( 43,135)( 44,136)( 45,137)( 46,138)( 47,131)( 48,132)( 49,133)( 50,134)
( 51,127)( 52,128)( 53,129)( 54,130)( 55,123)( 56,124)( 57,125)( 58,126)
( 59,147)( 60,148)( 61,149)( 62,150)( 63,143)( 64,144)( 65,145)( 66,146)
( 67,167)( 68,168)( 69,169)( 70,170)( 71,163)( 72,164)( 73,165)( 74,166)
( 75,159)( 76,160)( 77,161)( 78,162)( 79,155)( 80,156)( 81,157)( 82,158)
( 83,151)( 84,152)( 85,153)( 86,154);;
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, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(170)!(1,2);
s1 := 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);
s2 := Sym(170)!(  3, 31)(  4, 32)(  5, 34)(  6, 33)(  7, 35)(  8, 36)(  9, 38)
( 10, 37)( 11, 39)( 12, 40)( 13, 42)( 14, 41)( 15, 43)( 16, 44)( 17, 46)
( 18, 45)( 19, 47)( 20, 48)( 21, 50)( 22, 49)( 23, 51)( 24, 52)( 25, 54)
( 26, 53)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 61, 62)( 65, 66)( 69, 70)
( 73, 74)( 77, 78)( 81, 82)( 85, 86)( 87,115)( 88,116)( 89,118)( 90,117)
( 91,119)( 92,120)( 93,122)( 94,121)( 95,123)( 96,124)( 97,126)( 98,125)
( 99,127)(100,128)(101,130)(102,129)(103,131)(104,132)(105,134)(106,133)
(107,135)(108,136)(109,138)(110,137)(111,139)(112,140)(113,142)(114,141)
(145,146)(149,150)(153,154)(157,158)(161,162)(165,166)(169,170);
s3 := Sym(170)!(  3,  6)(  4,  5)(  7, 30)(  8, 29)(  9, 28)( 10, 27)( 11, 26)
( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 31, 34)
( 32, 33)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)
( 42, 51)( 43, 50)( 44, 49)( 45, 48)( 46, 47)( 59, 62)( 60, 61)( 63, 86)
( 64, 85)( 65, 84)( 66, 83)( 67, 82)( 68, 81)( 69, 80)( 70, 79)( 71, 78)
( 72, 77)( 73, 76)( 74, 75)( 87, 90)( 88, 89)( 91,114)( 92,113)( 93,112)
( 94,111)( 95,110)( 96,109)( 97,108)( 98,107)( 99,106)(100,105)(101,104)
(102,103)(115,118)(116,117)(119,142)(120,141)(121,140)(122,139)(123,138)
(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)(143,146)
(144,145)(147,170)(148,169)(149,168)(150,167)(151,166)(152,165)(153,164)
(154,163)(155,162)(156,161)(157,160)(158,159);
s4 := Sym(170)!(  3, 91)(  4, 92)(  5, 93)(  6, 94)(  7, 87)(  8, 88)(  9, 89)
( 10, 90)( 11,111)( 12,112)( 13,113)( 14,114)( 15,107)( 16,108)( 17,109)
( 18,110)( 19,103)( 20,104)( 21,105)( 22,106)( 23, 99)( 24,100)( 25,101)
( 26,102)( 27, 95)( 28, 96)( 29, 97)( 30, 98)( 31,119)( 32,120)( 33,121)
( 34,122)( 35,115)( 36,116)( 37,117)( 38,118)( 39,139)( 40,140)( 41,141)
( 42,142)( 43,135)( 44,136)( 45,137)( 46,138)( 47,131)( 48,132)( 49,133)
( 50,134)( 51,127)( 52,128)( 53,129)( 54,130)( 55,123)( 56,124)( 57,125)
( 58,126)( 59,147)( 60,148)( 61,149)( 62,150)( 63,143)( 64,144)( 65,145)
( 66,146)( 67,167)( 68,168)( 69,169)( 70,170)( 71,163)( 72,164)( 73,165)
( 74,166)( 75,159)( 76,160)( 77,161)( 78,162)( 79,155)( 80,156)( 81,157)
( 82,158)( 83,151)( 84,152)( 85,153)( 86,154);
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, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope