Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,8}

Atlas Canonical Name {6,8}*768h

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086333)
Rank
3
Schläfli Type
{6,8}
Vertices, edges, …
48, 192, 64
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*(s1*s2)^3*s1*s0*s2> of order 2

32 facets

32 vertex figures

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

32 facets

24 vertex figures

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

48 facets

24 vertex figures

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

32 facets

24 vertex figures

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

16 facets

16 vertex figures

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

16 facets

20 vertex figures

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

16 facets

12 vertex figures

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

16 facets

16 vertex figures

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

16 facets

16 vertex figures

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

16 facets

20 vertex figures

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

24 facets

16 vertex figures

P/N, where N=<(s1*s2)^4, s0*(s1*s0*s2)^2*s1*s2, s0*(s1*s2)^3*s1*s0*s2> of order 8

8 facets

10 vertex figures

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

8 facets

12 vertex figures

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

References

None.

to this polytope.

Twisty Puzzle