Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*768d

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086320)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
32, 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*s2)^2*s1)^2> of order 2

32 facets

16 vertex figures

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

32 facets

16 vertex figures

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

32 facets

16 vertex figures

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

40 facets

16 vertex figures

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

16 facets

8 vertex figures

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

24 facets

8 vertex figures

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

16 facets

8 vertex figures

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

16 facets

8 vertex figures

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

16 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle