Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,3}

Atlas Canonical Name {8,3}*384

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(384,5573)
Rank
3
Schläfli Type
{8,3}
Vertices, edges, …
64, 96, 24
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

32-fold

Covers minimal covers in bold

2-fold

3-fold

5-fold

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)^4> of order 2

16 facets

32 vertex figures

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

12 facets

32 vertex figures

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

12 facets

32 vertex figures

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

8 facets

16 vertex figures

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

10 facets

16 vertex figures

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

6 facets

16 vertex figures

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

10 facets

16 vertex figures

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

8 facets

16 vertex figures

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

6 facets

8 vertex figures

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

5 facets

8 vertex figures

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

4 facets

8 vertex figures

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

5 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle