Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,6}

Atlas Canonical Name {8,6}*768f

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086324)
Rank
3
Schläfli Type
{8,6}
Vertices, edges, …
64, 192, 48
Order of s0s1s2
12
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=<s1*s2*s1*s0*(s1*s2)^2*s1*s0*s1*s2> of order 2

24 facets

32 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=<(s1*s2)^3> of order 2

24 facets

40 vertex figures

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

24 facets

32 vertex figures

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

16 facets

16 vertex figures

P/N, where N=<s0*s2*s1*s0*s1*s2> of order 4

20 facets

16 vertex figures

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

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

12 facets

24 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^4*s2*s1*s0*s1*s2> of order 8

8 facets

8 vertex figures

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

10 facets

8 vertex figures

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

10 facets

8 vertex figures

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

12 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle