Polytope of Type {8,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,42}*672
Also Known As : {8,42|2}. if this polytope has another name.
Group : SmallGroup(672,975)
Rank : 3
Schlafli Type : {8,42}
Number of vertices, edges, etc : 8, 168, 42
Order of s₀s₁s₂ : 168
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,42,2} of size 1344
Vertex Figure Of :
   {2,8,42} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,42}*336a
   3-fold quotients : {8,14}*224
   4-fold quotients : {2,42}*168
   6-fold quotients : {4,14}*112
   7-fold quotients : {8,6}*96
   8-fold quotients : {2,21}*84
   12-fold quotients : {2,14}*56
   14-fold quotients : {4,6}*48a
   21-fold quotients : {8,2}*32
   24-fold quotients : {2,7}*28
   28-fold quotients : {2,6}*24
   42-fold quotients : {4,2}*16
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,84}*1344a, {16,42}*1344
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 >;

References : None.
to this polytope