Polytope of Type {2,4,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,42}*672c
if this polytope has a name.
Group : SmallGroup(672,1263)
Rank : 4
Schlafli Type : {2,4,42}
Number of vertices, edges, etc : 2, 4, 84, 42
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,42,2} of size 1344
Vertex Figure Of :
   {2,2,4,42} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,21}*336
   7-fold quotients : {2,4,6}*96b
   14-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,42}*1344
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope