Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,16}

Atlas Canonical Name {6,16}*768b

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Overview

Group
SmallGroup(768,1088585)
Rank
3
Schläfli Type
{6,16}
Vertices, edges, …
24, 192, 64
Order of s0s1s2
48
Order of s0s1s2s1
4
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

12-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*s2*s1)^2> of order 2

32 facets

12 vertex figures

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

32 facets

12 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^6*s2> of order 2

32 facets

12 vertex figures

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

32 facets

8 vertex figures

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

16 facets

6 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*(s2*s1)^2*s0*(s2*s1)^5*s2> of order 4

16 facets

6 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, s0*s1*s0*(s2*s1)^2*s0*(s2*s1)^5*s2> of order 4

16 facets

6 vertex figures

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

16 facets

6 vertex figures

Representations

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

References

None.

to this polytope.

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