Polytope of Type {2,8,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,42}*1344
if this polytope has a name.
Group : SmallGroup(1344,11133)
Rank : 4
Schlafli Type : {2,8,42}
Number of vertices, edges, etc : 2, 8, 168, 42
Order of s₀s₁s₂s₃ : 168
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 : {2,4,42}*672a
   3-fold quotients : {2,8,14}*448
   4-fold quotients : {2,2,42}*336
   6-fold quotients : {2,4,14}*224
   7-fold quotients : {2,8,6}*192
   8-fold quotients : {2,2,21}*168
   12-fold quotients : {2,2,14}*112
   14-fold quotients : {2,4,6}*96a
   21-fold quotients : {2,8,2}*64
   24-fold quotients : {2,2,7}*56
   28-fold quotients : {2,2,6}*48
   42-fold quotients : {2,4,2}*32
   56-fold quotients : {2,2,3}*24
   84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

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