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

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

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