Polytope of Type {2,42,8}

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

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

Permutation Representation (Magma) :

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

to this polytope