Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*768e

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1086324)
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=<((s1*s0)^2*s1*s2)^2> of order 2

32 facets

16 vertex figures

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

16 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=<s0*(s1*s0*s2)^3*s1*s2, (s0*s2*s1)^4> of order 4

16 facets

8 vertex figures

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

16 facets

8 vertex figures

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

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

References

None.

to this polytope.

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