Polytope of Type {2,42,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,42,6}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 4
Schlafli Type : {2,42,6}
Number of vertices, edges, etc : 2, 56, 168, 8
Order of s₀s₁s₂s₃ : 28
Order of s₀s₁s₂s₃s₂s₁ : 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) :
   2-fold quotients : {2,21,6}*672
   7-fold quotients : {2,6,6}*192
   12-fold quotients : {2,14,2}*112
   14-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,7,2}*56
   28-fold quotients : {2,3,3}*48
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope