Part of the Atlas of Small Regular Polytopes

Polytope of Type {168,2}

Atlas Canonical Name {168,2}*672

Overview

Group
SmallGroup(672,968)
Rank
3
Schläfli Type
{168,2}
Vertices, edges, …
168, 168, 2
Order of s0s1s2
168
Order of s0s1s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

7-fold

8-fold

12-fold

14-fold

21-fold

24-fold

28-fold

42-fold

56-fold

84-fold

Covers minimal covers in bold

2-fold

Representations

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