Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*768b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086329)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
64, 192, 64
Order of s0s1s2
8
Order of s0s1s2s1
12
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

32-fold

96-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*(s1*s2)^2*s1)^2> of order 2

32 facets

32 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*(s2*s1)^2*s0*s1*s2> of order 2

32 facets

32 vertex figures

P/N, where N=<(s0*s1)^3> of order 2

36 facets

32 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1> of order 2

32 facets

32 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1*s0*s1)^2*s2> of order 2

32 facets

36 vertex figures

P/N, where N=<(s0*s1)^2> of order 3

24 facets

24 vertex figures

P/N, where N=<s1*s0*(s1*s2)^2*s1*s0*s2*s1, s0*s1*s0*(s2*s1*s0*s1)^2*s2> of order 4

16 facets

20 vertex figures

P/N, where N=<(s1*s2*s1*s0)^2*s2*s1*s2> of order 4

16 facets

16 vertex figures

P/N, where N=<(s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1, s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1> of order 4

16 facets

16 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0, s0*s1*s0*(s2*s1*s0*s1)^2*s2> of order 4

16 facets

20 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1, s0*s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1> of order 4

16 facets

16 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1, (s0*(s1*s2)^2*s1)^2> of order 4

16 facets

16 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2> of order 4

20 facets

16 vertex figures

P/N, where N=<(s0*s1)^3, s1*s0*s2*s1*s0*(s2*s1)^2*s0*s1*s2> of order 4

20 facets

16 vertex figures

P/N, where N=<(s0*s1)^3, s1*s0*s2*(s1*s0)^2*s1*s2*s1, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0> of order 8

12 facets

8 vertex figures

P/N, where N=<s0*(s1*s2)^2*s1*s0*s2, (s0*s1)^2*s2*(s1*s0)^2*s1*s2*s1, (s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0> of order 8

8 facets

12 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 76)( 42, 75)( 43, 74)( 44, 73)( 45, 78)( 46, 77)( 47, 80)( 48, 79)( 49, 95)( 50, 96)( 51, 93)( 52, 94)( 53, 89)( 54, 90)( 55, 91)( 56, 92)( 57, 85)( 58, 86)( 59, 87)( 60, 88)( 61, 83)( 62, 84)( 63, 81)( 64, 82)( 97, 98)( 99,100)(101,104)(102,103)(105,107)(106,108)(113,128)(114,127)(115,126)(116,125)(117,122)(118,121)(119,124)(120,123)(129,162)(130,161)(131,164)(132,163)(133,168)(134,167)(135,166)(136,165)(137,171)(138,172)(139,169)(140,170)(141,173)(142,174)(143,175)(144,176)(145,192)(146,191)(147,190)(148,189)(149,186)(150,185)(151,188)(152,187)(153,182)(154,181)(155,184)(156,183)(157,180)(158,179)(159,178)(160,177);;
s1 := (  1, 65)(  2, 66)(  3, 69)(  4, 70)(  5, 67)(  6, 68)(  7, 71)(  8, 72)(  9, 82)( 10, 81)( 11, 86)( 12, 85)( 13, 84)( 14, 83)( 15, 88)( 16, 87)( 17, 74)( 18, 73)( 19, 78)( 20, 77)( 21, 76)( 22, 75)( 23, 80)( 24, 79)( 25, 90)( 26, 89)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)( 32, 95)( 35, 37)( 36, 38)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 57, 58)( 59, 62)( 60, 61)( 63, 64)( 97,161)( 98,162)( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,178)(106,177)(107,182)(108,181)(109,180)(110,179)(111,184)(112,183)(113,170)(114,169)(115,174)(116,173)(117,172)(118,171)(119,176)(120,175)(121,186)(122,185)(123,190)(124,189)(125,188)(126,187)(127,192)(128,191)(131,133)(132,134)(137,146)(138,145)(139,150)(140,149)(141,148)(142,147)(143,152)(144,151)(153,154)(155,158)(156,157)(159,160);;
s2 := (  1,112)(  2,111)(  3,110)(  4,109)(  5,106)(  6,105)(  7,108)(  8,107)(  9,102)( 10,101)( 11,104)( 12,103)( 13,100)( 14, 99)( 15, 98)( 16, 97)( 17,113)( 18,114)( 19,115)( 20,116)( 21,119)( 22,120)( 23,117)( 24,118)( 25,124)( 26,123)( 27,122)( 28,121)( 29,126)( 30,125)( 31,128)( 32,127)( 33,176)( 34,175)( 35,174)( 36,173)( 37,170)( 38,169)( 39,172)( 40,171)( 41,166)( 42,165)( 43,168)( 44,167)( 45,164)( 46,163)( 47,162)( 48,161)( 49,177)( 50,178)( 51,179)( 52,180)( 53,183)( 54,184)( 55,181)( 56,182)( 57,188)( 58,187)( 59,186)( 60,185)( 61,190)( 62,189)( 63,192)( 64,191)( 65,144)( 66,143)( 67,142)( 68,141)( 69,138)( 70,137)( 71,140)( 72,139)( 73,134)( 74,133)( 75,136)( 76,135)( 77,132)( 78,131)( 79,130)( 80,129)( 81,145)( 82,146)( 83,147)( 84,148)( 85,151)( 86,152)( 87,149)( 88,150)( 89,156)( 90,155)( 91,154)( 92,153)( 93,158)( 94,157)( 95,160)( 96,159);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(192)!(  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 76)( 42, 75)( 43, 74)( 44, 73)( 45, 78)( 46, 77)( 47, 80)( 48, 79)( 49, 95)( 50, 96)( 51, 93)( 52, 94)( 53, 89)( 54, 90)( 55, 91)( 56, 92)( 57, 85)( 58, 86)( 59, 87)( 60, 88)( 61, 83)( 62, 84)( 63, 81)( 64, 82)( 97, 98)( 99,100)(101,104)(102,103)(105,107)(106,108)(113,128)(114,127)(115,126)(116,125)(117,122)(118,121)(119,124)(120,123)(129,162)(130,161)(131,164)(132,163)(133,168)(134,167)(135,166)(136,165)(137,171)(138,172)(139,169)(140,170)(141,173)(142,174)(143,175)(144,176)(145,192)(146,191)(147,190)(148,189)(149,186)(150,185)(151,188)(152,187)(153,182)(154,181)(155,184)(156,183)(157,180)(158,179)(159,178)(160,177);
s1 := Sym(192)!(  1, 65)(  2, 66)(  3, 69)(  4, 70)(  5, 67)(  6, 68)(  7, 71)(  8, 72)(  9, 82)( 10, 81)( 11, 86)( 12, 85)( 13, 84)( 14, 83)( 15, 88)( 16, 87)( 17, 74)( 18, 73)( 19, 78)( 20, 77)( 21, 76)( 22, 75)( 23, 80)( 24, 79)( 25, 90)( 26, 89)( 27, 94)( 28, 93)( 29, 92)( 30, 91)( 31, 96)( 32, 95)( 35, 37)( 36, 38)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 57, 58)( 59, 62)( 60, 61)( 63, 64)( 97,161)( 98,162)( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,178)(106,177)(107,182)(108,181)(109,180)(110,179)(111,184)(112,183)(113,170)(114,169)(115,174)(116,173)(117,172)(118,171)(119,176)(120,175)(121,186)(122,185)(123,190)(124,189)(125,188)(126,187)(127,192)(128,191)(131,133)(132,134)(137,146)(138,145)(139,150)(140,149)(141,148)(142,147)(143,152)(144,151)(153,154)(155,158)(156,157)(159,160);
s2 := Sym(192)!(  1,112)(  2,111)(  3,110)(  4,109)(  5,106)(  6,105)(  7,108)(  8,107)(  9,102)( 10,101)( 11,104)( 12,103)( 13,100)( 14, 99)( 15, 98)( 16, 97)( 17,113)( 18,114)( 19,115)( 20,116)( 21,119)( 22,120)( 23,117)( 24,118)( 25,124)( 26,123)( 27,122)( 28,121)( 29,126)( 30,125)( 31,128)( 32,127)( 33,176)( 34,175)( 35,174)( 36,173)( 37,170)( 38,169)( 39,172)( 40,171)( 41,166)( 42,165)( 43,168)( 44,167)( 45,164)( 46,163)( 47,162)( 48,161)( 49,177)( 50,178)( 51,179)( 52,180)( 53,183)( 54,184)( 55,181)( 56,182)( 57,188)( 58,187)( 59,186)( 60,185)( 61,190)( 62,189)( 63,192)( 64,191)( 65,144)( 66,143)( 67,142)( 68,141)( 69,138)( 70,137)( 71,140)( 72,139)( 73,134)( 74,133)( 75,136)( 76,135)( 77,132)( 78,131)( 79,130)( 80,129)( 81,145)( 82,146)( 83,147)( 84,148)( 85,151)( 86,152)( 87,149)( 88,150)( 89,156)( 90,155)( 91,154)( 92,153)( 93,158)( 94,157)( 95,160)( 96,159);
poly := sub<Sym(192)|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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;

References

None.

to this polytope.

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