Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*768c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086333)
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

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*s0*s2)^3*s1*s2> of order 2

32 facets

32 vertex figures

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

32 facets

32 vertex figures

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

48 facets

32 vertex figures

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

32 facets

32 vertex figures

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

16 facets

16 vertex figures

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

16 facets

16 vertex figures

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

24 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*s2*s1*s0*(s2*s1)^2> of order 4

16 facets

20 vertex figures

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

16 facets

16 vertex figures

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

16 facets

16 vertex figures

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

16 facets

20 vertex figures

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

16 facets

16 vertex figures

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

16 facets

16 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)^3, s0*(s1*s0*s2)^3*s1*s2> of order 4

24 facets

16 vertex figures

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

8 facets

12 vertex figures

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

12 facets

8 vertex figures

P/N, where N=<(s0*s1)^3, s0*(s1*s0*s2)^3*s1*s2, (s0*s2*s1)^4> of order 8

12 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle