Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,12}

Atlas Canonical Name {8,12}*768t

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Overview

Group
SmallGroup(768,1087745)
Rank
3
Schläfli Type
{8,12}
Vertices, edges, …
32, 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

24-fold

32-fold

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

24 facets

16 vertex figures

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

24 facets

16 vertex figures

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

24 facets

16 vertex figures

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

16 facets

16 vertex figures

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

12 facets

8 vertex figures

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

12 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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