Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,6}

Atlas Canonical Name {8,6}*768h

▶ 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)^2*s0*(s2*s1)^2*s0*(s1*s2)^2> of order 2

24 facets

32 vertex figures

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

24 facets

48 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*s0)^3*s1*s2, (s0*s1)^2*(s2*s1*s0)^2*s2*s1> of order 4

16 facets

16 vertex figures

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

20 facets

16 vertex figures

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

16 facets

16 vertex figures

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

16 facets

16 vertex figures

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

20 facets

16 vertex figures

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

16 facets

24 vertex figures

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

10 facets

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

8 facets

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

References

None.

to this polytope.

Twisty Puzzle