Part of the Atlas of Small Regular Polytopes

Polytope of Type {164,6}

Atlas Canonical Name {164,6}*1968b

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Overview

Group
SmallGroup(1968,188)
Rank
3
Schläfli Type
{164,6}
Vertices, edges, …
164, 492, 6
Order of s0s1s2
123
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

41-fold

82-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

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