Polytope of Type {2,168}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,168}*672
if this polytope has a name.
Group : SmallGroup(672,968)
Rank : 3
Schlafli Type : {2,168}
Number of vertices, edges, etc : 2, 168, 168
Order of s₀s₁s₂ : 168
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,168,2} of size 1344
Vertex Figure Of :
   {2,2,168} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,84}*336
   3-fold quotients : {2,56}*224
   4-fold quotients : {2,42}*168
   6-fold quotients : {2,28}*112
   7-fold quotients : {2,24}*96
   8-fold quotients : {2,21}*84
   12-fold quotients : {2,14}*56
   14-fold quotients : {2,12}*48
   21-fold quotients : {2,8}*32
   24-fold quotients : {2,7}*28
   28-fold quotients : {2,6}*24
   42-fold quotients : {2,4}*16
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,168}*1344a, {2,336}*1344
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)( 45, 66)( 46, 72)( 47, 71)( 48, 70)
( 49, 69)( 50, 68)( 51, 67)( 52, 80)( 53, 86)( 54, 85)( 55, 84)( 56, 83)
( 57, 82)( 58, 81)( 59, 73)( 60, 79)( 61, 78)( 62, 77)( 63, 76)( 64, 75)
( 65, 74)( 87,129)( 88,135)( 89,134)( 90,133)( 91,132)( 92,131)( 93,130)
( 94,143)( 95,149)( 96,148)( 97,147)( 98,146)( 99,145)(100,144)(101,136)
(102,142)(103,141)(104,140)(105,139)(106,138)(107,137)(108,150)(109,156)
(110,155)(111,154)(112,153)(113,152)(114,151)(115,164)(116,170)(117,169)
(118,168)(119,167)(120,166)(121,165)(122,157)(123,163)(124,162)(125,161)
(126,160)(127,159)(128,158);;
s2 := (  3, 95)(  4, 94)(  5,100)(  6, 99)(  7, 98)(  8, 97)(  9, 96)( 10, 88)
( 11, 87)( 12, 93)( 13, 92)( 14, 91)( 15, 90)( 16, 89)( 17,102)( 18,101)
( 19,107)( 20,106)( 21,105)( 22,104)( 23,103)( 24,116)( 25,115)( 26,121)
( 27,120)( 28,119)( 29,118)( 30,117)( 31,109)( 32,108)( 33,114)( 34,113)
( 35,112)( 36,111)( 37,110)( 38,123)( 39,122)( 40,128)( 41,127)( 42,126)
( 43,125)( 44,124)( 45,158)( 46,157)( 47,163)( 48,162)( 49,161)( 50,160)
( 51,159)( 52,151)( 53,150)( 54,156)( 55,155)( 56,154)( 57,153)( 58,152)
( 59,165)( 60,164)( 61,170)( 62,169)( 63,168)( 64,167)( 65,166)( 66,137)
( 67,136)( 68,142)( 69,141)( 70,140)( 71,139)( 72,138)( 73,130)( 74,129)
( 75,135)( 76,134)( 77,133)( 78,132)( 79,131)( 80,144)( 81,143)( 82,149)
( 83,148)( 84,147)( 85,146)( 86,145);;
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*s1*s0*s1, s0*s2*s0*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*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*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*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*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)( 45, 66)( 46, 72)( 47, 71)
( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 80)( 53, 86)( 54, 85)( 55, 84)
( 56, 83)( 57, 82)( 58, 81)( 59, 73)( 60, 79)( 61, 78)( 62, 77)( 63, 76)
( 64, 75)( 65, 74)( 87,129)( 88,135)( 89,134)( 90,133)( 91,132)( 92,131)
( 93,130)( 94,143)( 95,149)( 96,148)( 97,147)( 98,146)( 99,145)(100,144)
(101,136)(102,142)(103,141)(104,140)(105,139)(106,138)(107,137)(108,150)
(109,156)(110,155)(111,154)(112,153)(113,152)(114,151)(115,164)(116,170)
(117,169)(118,168)(119,167)(120,166)(121,165)(122,157)(123,163)(124,162)
(125,161)(126,160)(127,159)(128,158);
s2 := Sym(170)!(  3, 95)(  4, 94)(  5,100)(  6, 99)(  7, 98)(  8, 97)(  9, 96)
( 10, 88)( 11, 87)( 12, 93)( 13, 92)( 14, 91)( 15, 90)( 16, 89)( 17,102)
( 18,101)( 19,107)( 20,106)( 21,105)( 22,104)( 23,103)( 24,116)( 25,115)
( 26,121)( 27,120)( 28,119)( 29,118)( 30,117)( 31,109)( 32,108)( 33,114)
( 34,113)( 35,112)( 36,111)( 37,110)( 38,123)( 39,122)( 40,128)( 41,127)
( 42,126)( 43,125)( 44,124)( 45,158)( 46,157)( 47,163)( 48,162)( 49,161)
( 50,160)( 51,159)( 52,151)( 53,150)( 54,156)( 55,155)( 56,154)( 57,153)
( 58,152)( 59,165)( 60,164)( 61,170)( 62,169)( 63,168)( 64,167)( 65,166)
( 66,137)( 67,136)( 68,142)( 69,141)( 70,140)( 71,139)( 72,138)( 73,130)
( 74,129)( 75,135)( 76,134)( 77,133)( 78,132)( 79,131)( 80,144)( 81,143)
( 82,149)( 83,148)( 84,147)( 85,146)( 86,145);
poly := sub<Sym(170)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

to this polytope