Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,164}

Atlas Canonical Name {6,164}*1968b

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Overview

Group
SmallGroup(1968,188)
Rank
3
Schläfli Type
{6,164}
Vertices, edges, …
6, 492, 164
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 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 83, 84)( 87, 88)( 91, 92)( 95, 96)( 99,100)(103,104)(107,108)(111,112)(115,116)(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(147,148)(151,152)(155,156)(159,160)(163,164);;
s1 := (  2,  4)(  5,161)(  6,164)(  7,163)(  8,162)(  9,157)( 10,160)( 11,159)( 12,158)( 13,153)( 14,156)( 15,155)( 16,154)( 17,149)( 18,152)( 19,151)( 20,150)( 21,145)( 22,148)( 23,147)( 24,146)( 25,141)( 26,144)( 27,143)( 28,142)( 29,137)( 30,140)( 31,139)( 32,138)( 33,133)( 34,136)( 35,135)( 36,134)( 37,129)( 38,132)( 39,131)( 40,130)( 41,125)( 42,128)( 43,127)( 44,126)( 45,121)( 46,124)( 47,123)( 48,122)( 49,117)( 50,120)( 51,119)( 52,118)( 53,113)( 54,116)( 55,115)( 56,114)( 57,109)( 58,112)( 59,111)( 60,110)( 61,105)( 62,108)( 63,107)( 64,106)( 65,101)( 66,104)( 67,103)( 68,102)( 69, 97)( 70,100)( 71, 99)( 72, 98)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 85)( 82, 88)( 83, 87)( 84, 86);;
s2 := (  1,  6)(  2,  5)(  3,  8)(  4,  7)(  9,162)( 10,161)( 11,164)( 12,163)( 13,158)( 14,157)( 15,160)( 16,159)( 17,154)( 18,153)( 19,156)( 20,155)( 21,150)( 22,149)( 23,152)( 24,151)( 25,146)( 26,145)( 27,148)( 28,147)( 29,142)( 30,141)( 31,144)( 32,143)( 33,138)( 34,137)( 35,140)( 36,139)( 37,134)( 38,133)( 39,136)( 40,135)( 41,130)( 42,129)( 43,132)( 44,131)( 45,126)( 46,125)( 47,128)( 48,127)( 49,122)( 50,121)( 51,124)( 52,123)( 53,118)( 54,117)( 55,120)( 56,119)( 57,114)( 58,113)( 59,116)( 60,115)( 61,110)( 62,109)( 63,112)( 64,111)( 65,106)( 66,105)( 67,108)( 68,107)( 69,102)( 70,101)( 71,104)( 72,103)( 73, 98)( 74, 97)( 75,100)( 76, 99)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 81, 90)( 82, 89)( 83, 92)( 84, 91)( 85, 86)( 87, 88);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(164)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 83, 84)( 87, 88)( 91, 92)( 95, 96)( 99,100)(103,104)(107,108)(111,112)(115,116)(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(147,148)(151,152)(155,156)(159,160)(163,164);
s1 := Sym(164)!(  2,  4)(  5,161)(  6,164)(  7,163)(  8,162)(  9,157)( 10,160)( 11,159)( 12,158)( 13,153)( 14,156)( 15,155)( 16,154)( 17,149)( 18,152)( 19,151)( 20,150)( 21,145)( 22,148)( 23,147)( 24,146)( 25,141)( 26,144)( 27,143)( 28,142)( 29,137)( 30,140)( 31,139)( 32,138)( 33,133)( 34,136)( 35,135)( 36,134)( 37,129)( 38,132)( 39,131)( 40,130)( 41,125)( 42,128)( 43,127)( 44,126)( 45,121)( 46,124)( 47,123)( 48,122)( 49,117)( 50,120)( 51,119)( 52,118)( 53,113)( 54,116)( 55,115)( 56,114)( 57,109)( 58,112)( 59,111)( 60,110)( 61,105)( 62,108)( 63,107)( 64,106)( 65,101)( 66,104)( 67,103)( 68,102)( 69, 97)( 70,100)( 71, 99)( 72, 98)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 85)( 82, 88)( 83, 87)( 84, 86);
s2 := Sym(164)!(  1,  6)(  2,  5)(  3,  8)(  4,  7)(  9,162)( 10,161)( 11,164)( 12,163)( 13,158)( 14,157)( 15,160)( 16,159)( 17,154)( 18,153)( 19,156)( 20,155)( 21,150)( 22,149)( 23,152)( 24,151)( 25,146)( 26,145)( 27,148)( 28,147)( 29,142)( 30,141)( 31,144)( 32,143)( 33,138)( 34,137)( 35,140)( 36,139)( 37,134)( 38,133)( 39,136)( 40,135)( 41,130)( 42,129)( 43,132)( 44,131)( 45,126)( 46,125)( 47,128)( 48,127)( 49,122)( 50,121)( 51,124)( 52,123)( 53,118)( 54,117)( 55,120)( 56,119)( 57,114)( 58,113)( 59,116)( 60,115)( 61,110)( 62,109)( 63,112)( 64,111)( 65,106)( 66,105)( 67,108)( 68,107)( 69,102)( 70,101)( 71,104)( 72,103)( 73, 98)( 74, 97)( 75,100)( 76, 99)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 81, 90)( 82, 89)( 83, 92)( 84, 91)( 85, 86)( 87, 88);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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.

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