Polytope of Type {2,2,42,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,42,4}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11701)
Rank : 5
Schlafli Type : {2,2,42,4}
Number of vertices, edges, etc : 2, 2, 42, 84, 4
Order of s₀s₁s₂s₃s₄ : 42
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,2,21,4}*672
   7-fold quotients : {2,2,6,4}*192c
   14-fold quotients : {2,2,3,4}*96
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := (  6,  7)(  9, 29)( 10, 31)( 11, 30)( 12, 32)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 33, 61)( 34, 63)( 35, 62)
( 36, 64)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 81)( 42, 83)( 43, 82)
( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)( 59, 66)
( 60, 68)( 90, 91)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)( 98,111)
( 99,110)(100,112)(101,105)(102,107)(103,106)(104,108)(117,145)(118,147)
(119,146)(120,148)(121,169)(122,171)(123,170)(124,172)(125,165)(126,167)
(127,166)(128,168)(129,161)(130,163)(131,162)(132,164)(133,157)(134,159)
(135,158)(136,160)(137,153)(138,155)(139,154)(140,156)(141,149)(142,151)
(143,150)(144,152);;
s3 := (  5,121)(  6,122)(  7,124)(  8,123)(  9,117)( 10,118)( 11,120)( 12,119)
( 13,141)( 14,142)( 15,144)( 16,143)( 17,137)( 18,138)( 19,140)( 20,139)
( 21,133)( 22,134)( 23,136)( 24,135)( 25,129)( 26,130)( 27,132)( 28,131)
( 29,125)( 30,126)( 31,128)( 32,127)( 33, 93)( 34, 94)( 35, 96)( 36, 95)
( 37, 89)( 38, 90)( 39, 92)( 40, 91)( 41,113)( 42,114)( 43,116)( 44,115)
( 45,109)( 46,110)( 47,112)( 48,111)( 49,105)( 50,106)( 51,108)( 52,107)
( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)( 60, 99)
( 61,149)( 62,150)( 63,152)( 64,151)( 65,145)( 66,146)( 67,148)( 68,147)
( 69,169)( 70,170)( 71,172)( 72,171)( 73,165)( 74,166)( 75,168)( 76,167)
( 77,161)( 78,162)( 79,164)( 80,163)( 81,157)( 82,158)( 83,160)( 84,159)
( 85,153)( 86,154)( 87,156)( 88,155);;
s4 := (  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)( 18, 19)
( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)( 34, 35)
( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 52)( 50, 51)
( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)( 66, 67)
( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 84)( 82, 83)
( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)( 97,100)( 98, 99)
(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)(113,116)(114,115)
(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)(130,131)
(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)(145,148)(146,147)
(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)(161,164)(162,163)
(165,168)(166,167)(169,172)(170,171);;
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, s1*s2*s1*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*s3*s4, 
s4*s3*s2*s4*s3*s4*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*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*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*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(172)!(1,2);
s1 := Sym(172)!(3,4);
s2 := Sym(172)!(  6,  7)(  9, 29)( 10, 31)( 11, 30)( 12, 32)( 13, 25)( 14, 27)
( 15, 26)( 16, 28)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 33, 61)( 34, 63)
( 35, 62)( 36, 64)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 81)( 42, 83)
( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 73)( 50, 75)
( 51, 74)( 52, 76)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)
( 59, 66)( 60, 68)( 90, 91)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)
( 98,111)( 99,110)(100,112)(101,105)(102,107)(103,106)(104,108)(117,145)
(118,147)(119,146)(120,148)(121,169)(122,171)(123,170)(124,172)(125,165)
(126,167)(127,166)(128,168)(129,161)(130,163)(131,162)(132,164)(133,157)
(134,159)(135,158)(136,160)(137,153)(138,155)(139,154)(140,156)(141,149)
(142,151)(143,150)(144,152);
s3 := Sym(172)!(  5,121)(  6,122)(  7,124)(  8,123)(  9,117)( 10,118)( 11,120)
( 12,119)( 13,141)( 14,142)( 15,144)( 16,143)( 17,137)( 18,138)( 19,140)
( 20,139)( 21,133)( 22,134)( 23,136)( 24,135)( 25,129)( 26,130)( 27,132)
( 28,131)( 29,125)( 30,126)( 31,128)( 32,127)( 33, 93)( 34, 94)( 35, 96)
( 36, 95)( 37, 89)( 38, 90)( 39, 92)( 40, 91)( 41,113)( 42,114)( 43,116)
( 44,115)( 45,109)( 46,110)( 47,112)( 48,111)( 49,105)( 50,106)( 51,108)
( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)
( 60, 99)( 61,149)( 62,150)( 63,152)( 64,151)( 65,145)( 66,146)( 67,148)
( 68,147)( 69,169)( 70,170)( 71,172)( 72,171)( 73,165)( 74,166)( 75,168)
( 76,167)( 77,161)( 78,162)( 79,164)( 80,163)( 81,157)( 82,158)( 83,160)
( 84,159)( 85,153)( 86,154)( 87,156)( 88,155);
s4 := Sym(172)!(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 20)
( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)( 33, 36)
( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 52)
( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)( 65, 68)
( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 84)
( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)( 97,100)
( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)(113,116)
(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)(129,132)
(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)(145,148)
(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)(161,164)
(162,163)(165,168)(166,167)(169,172)(170,171);
poly := sub<Sym(172)|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, 
s1*s2*s1*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*s3*s4, s4*s3*s2*s4*s3*s4*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*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*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*s3*s2*s3 >;

to this polytope