Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,6}

Atlas Canonical Name {8,6}*768i

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086333)
Rank
3
Schläfli Type
{8,6}
Vertices, edges, …
64, 192, 48
Order of s0s1s2
6
Order of s0s1s2s1
8
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

64-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*s2*(s1*s0)^3*s1*s2> of order 2

32 facets

32 vertex figures

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

24 facets

32 vertex figures

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

24 facets

32 vertex figures

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

24 facets

32 vertex figures

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

12 facets

16 vertex figures

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

20 facets

16 vertex figures

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

12 facets

16 vertex figures

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

16 facets

16 vertex figures

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

16 facets

16 vertex figures

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

20 facets

16 vertex figures

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

16 facets

16 vertex figures

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

12 facets

8 vertex figures

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

10 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle