Polytope of Type {2,6,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,42}*1344
if this polytope has a name.
Group : SmallGroup(1344,11695)
Rank : 4
Schlafli Type : {2,6,42}
Number of vertices, edges, etc : 2, 8, 168, 56
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 : {2,6,21}*672
   7-fold quotients : {2,6,6}*192
   12-fold quotients : {2,2,14}*112
   14-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,2,7}*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)(  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, 55)(  8, 56)(  9, 58)( 10, 57)
( 11, 51)( 12, 52)( 13, 54)( 14, 53)( 15, 47)( 16, 48)( 17, 50)( 18, 49)
( 19, 43)( 20, 44)( 21, 46)( 22, 45)( 23, 39)( 24, 40)( 25, 42)( 26, 41)
( 27, 35)( 28, 36)( 29, 38)( 30, 37)( 61, 62)( 63, 83)( 64, 84)( 65, 86)
( 66, 85)( 67, 79)( 68, 80)( 69, 82)( 70, 81)( 71, 75)( 72, 76)( 73, 78)
( 74, 77)( 87,115)( 88,116)( 89,118)( 90,117)( 91,139)( 92,140)( 93,142)
( 94,141)( 95,135)( 96,136)( 97,138)( 98,137)( 99,131)(100,132)(101,134)
(102,133)(103,127)(104,128)(105,130)(106,129)(107,123)(108,124)(109,126)
(110,125)(111,119)(112,120)(113,122)(114,121)(145,146)(147,167)(148,168)
(149,170)(150,169)(151,163)(152,164)(153,166)(154,165)(155,159)(156,160)
(157,162)(158,161);;
s3 := (  3, 94)(  4, 92)(  5, 93)(  6, 91)(  7, 90)(  8, 88)(  9, 89)( 10, 87)
( 11,114)( 12,112)( 13,113)( 14,111)( 15,110)( 16,108)( 17,109)( 18,107)
( 19,106)( 20,104)( 21,105)( 22,103)( 23,102)( 24,100)( 25,101)( 26, 99)
( 27, 98)( 28, 96)( 29, 97)( 30, 95)( 31,150)( 32,148)( 33,149)( 34,147)
( 35,146)( 36,144)( 37,145)( 38,143)( 39,170)( 40,168)( 41,169)( 42,167)
( 43,166)( 44,164)( 45,165)( 46,163)( 47,162)( 48,160)( 49,161)( 50,159)
( 51,158)( 52,156)( 53,157)( 54,155)( 55,154)( 56,152)( 57,153)( 58,151)
( 59,122)( 60,120)( 61,121)( 62,119)( 63,118)( 64,116)( 65,117)( 66,115)
( 67,142)( 68,140)( 69,141)( 70,139)( 71,138)( 72,136)( 73,137)( 74,135)
( 75,134)( 76,132)( 77,133)( 78,131)( 79,130)( 80,128)( 81,129)( 82,127)
( 83,126)( 84,124)( 85,125)( 86,123);;
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*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3 ];;
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, 55)(  8, 56)(  9, 58)
( 10, 57)( 11, 51)( 12, 52)( 13, 54)( 14, 53)( 15, 47)( 16, 48)( 17, 50)
( 18, 49)( 19, 43)( 20, 44)( 21, 46)( 22, 45)( 23, 39)( 24, 40)( 25, 42)
( 26, 41)( 27, 35)( 28, 36)( 29, 38)( 30, 37)( 61, 62)( 63, 83)( 64, 84)
( 65, 86)( 66, 85)( 67, 79)( 68, 80)( 69, 82)( 70, 81)( 71, 75)( 72, 76)
( 73, 78)( 74, 77)( 87,115)( 88,116)( 89,118)( 90,117)( 91,139)( 92,140)
( 93,142)( 94,141)( 95,135)( 96,136)( 97,138)( 98,137)( 99,131)(100,132)
(101,134)(102,133)(103,127)(104,128)(105,130)(106,129)(107,123)(108,124)
(109,126)(110,125)(111,119)(112,120)(113,122)(114,121)(145,146)(147,167)
(148,168)(149,170)(150,169)(151,163)(152,164)(153,166)(154,165)(155,159)
(156,160)(157,162)(158,161);
s3 := Sym(170)!(  3, 94)(  4, 92)(  5, 93)(  6, 91)(  7, 90)(  8, 88)(  9, 89)
( 10, 87)( 11,114)( 12,112)( 13,113)( 14,111)( 15,110)( 16,108)( 17,109)
( 18,107)( 19,106)( 20,104)( 21,105)( 22,103)( 23,102)( 24,100)( 25,101)
( 26, 99)( 27, 98)( 28, 96)( 29, 97)( 30, 95)( 31,150)( 32,148)( 33,149)
( 34,147)( 35,146)( 36,144)( 37,145)( 38,143)( 39,170)( 40,168)( 41,169)
( 42,167)( 43,166)( 44,164)( 45,165)( 46,163)( 47,162)( 48,160)( 49,161)
( 50,159)( 51,158)( 52,156)( 53,157)( 54,155)( 55,154)( 56,152)( 57,153)
( 58,151)( 59,122)( 60,120)( 61,121)( 62,119)( 63,118)( 64,116)( 65,117)
( 66,115)( 67,142)( 68,140)( 69,141)( 70,139)( 71,138)( 72,136)( 73,137)
( 74,135)( 75,134)( 76,132)( 77,133)( 78,131)( 79,130)( 80,128)( 81,129)
( 82,127)( 83,126)( 84,124)( 85,125)( 86,123);
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*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3 >;

to this polytope