Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,12}

Atlas Canonical Name {4,12}*864c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(864,4080)
Rank
3
Schläfli Type
{4,12}
Vertices, edges, …
36, 216, 108
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

6-fold

9-fold

12-fold

18-fold

27-fold

36-fold

54-fold

72-fold

108-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

54 facets

18 vertex figures

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

60 facets

18 vertex figures

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

36 facets

12 vertex figures

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

36 facets

18 vertex figures

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

36 facets

12 vertex figures

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

36 facets

12 vertex figures

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

24 facets

6 vertex figures

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

24 facets

6 vertex figures

P/N, where N=<(s1*s2)^4, s0*s1*s0*(s2*s1)^2*s0*s2*s1*s0*s2> of order 9

12 facets

6 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, s1*s0*(s1*s2)^3*s1*s0*s2*s1> of order 9

12 facets

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  4, 22)(  5, 23)(  6, 24)(  7, 16)(  8, 17)(  9, 18)( 10, 19)( 11, 20)( 12, 21)( 31, 49)( 32, 50)( 33, 51)( 34, 43)( 35, 44)( 36, 45)( 37, 46)( 38, 47)( 39, 48)( 58, 76)( 59, 77)( 60, 78)( 61, 70)( 62, 71)( 63, 72)( 64, 73)( 65, 74)( 66, 75)( 85,103)( 86,104)( 87,105)( 88, 97)( 89, 98)( 90, 99)( 91,100)( 92,101)( 93,102)(109,163)(110,164)(111,165)(112,184)(113,185)(114,186)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,175)(122,176)(123,177)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,166)(131,167)(132,168)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,211)(140,212)(141,213)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,202)(149,203)(150,204)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,193)(158,194)(159,195)(160,214)(161,215)(162,216);;
s1 := (  1,109)(  2,111)(  3,110)(  4,112)(  5,114)(  6,113)(  7,115)(  8,117)(  9,116)( 10,133)( 11,135)( 12,134)( 13,127)( 14,129)( 15,128)( 16,130)( 17,132)( 18,131)( 19,121)( 20,123)( 21,122)( 22,124)( 23,126)( 24,125)( 25,118)( 26,120)( 27,119)( 28,136)( 29,138)( 30,137)( 31,139)( 32,141)( 33,140)( 34,142)( 35,144)( 36,143)( 37,160)( 38,162)( 39,161)( 40,154)( 41,156)( 42,155)( 43,157)( 44,159)( 45,158)( 46,148)( 47,150)( 48,149)( 49,151)( 50,153)( 51,152)( 52,145)( 53,147)( 54,146)( 55,163)( 56,165)( 57,164)( 58,166)( 59,168)( 60,167)( 61,169)( 62,171)( 63,170)( 64,187)( 65,189)( 66,188)( 67,181)( 68,183)( 69,182)( 70,184)( 71,186)( 72,185)( 73,175)( 74,177)( 75,176)( 76,178)( 77,180)( 78,179)( 79,172)( 80,174)( 81,173)( 82,190)( 83,192)( 84,191)( 85,193)( 86,195)( 87,194)( 88,196)( 89,198)( 90,197)( 91,214)( 92,216)( 93,215)( 94,208)( 95,210)( 96,209)( 97,211)( 98,213)( 99,212)(100,202)(101,204)(102,203)(103,205)(104,207)(105,206)(106,199)(107,201)(108,200);;
s2 := (  1, 68)(  2, 67)(  3, 69)(  4, 65)(  5, 64)(  6, 66)(  7, 71)(  8, 70)(  9, 72)( 10, 59)( 11, 58)( 12, 60)( 13, 56)( 14, 55)( 15, 57)( 16, 62)( 17, 61)( 18, 63)( 19, 77)( 20, 76)( 21, 78)( 22, 74)( 23, 73)( 24, 75)( 25, 80)( 26, 79)( 27, 81)( 28, 95)( 29, 94)( 30, 96)( 31, 92)( 32, 91)( 33, 93)( 34, 98)( 35, 97)( 36, 99)( 37, 86)( 38, 85)( 39, 87)( 40, 83)( 41, 82)( 42, 84)( 43, 89)( 44, 88)( 45, 90)( 46,104)( 47,103)( 48,105)( 49,101)( 50,100)( 51,102)( 52,107)( 53,106)( 54,108)(109,203)(110,202)(111,204)(112,200)(113,199)(114,201)(115,206)(116,205)(117,207)(118,194)(119,193)(120,195)(121,191)(122,190)(123,192)(124,197)(125,196)(126,198)(127,212)(128,211)(129,213)(130,209)(131,208)(132,210)(133,215)(134,214)(135,216)(136,176)(137,175)(138,177)(139,173)(140,172)(141,174)(142,179)(143,178)(144,180)(145,167)(146,166)(147,168)(148,164)(149,163)(150,165)(151,170)(152,169)(153,171)(154,185)(155,184)(156,186)(157,182)(158,181)(159,183)(160,188)(161,187)(162,189);;
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*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(216)!(  4, 22)(  5, 23)(  6, 24)(  7, 16)(  8, 17)(  9, 18)( 10, 19)( 11, 20)( 12, 21)( 31, 49)( 32, 50)( 33, 51)( 34, 43)( 35, 44)( 36, 45)( 37, 46)( 38, 47)( 39, 48)( 58, 76)( 59, 77)( 60, 78)( 61, 70)( 62, 71)( 63, 72)( 64, 73)( 65, 74)( 66, 75)( 85,103)( 86,104)( 87,105)( 88, 97)( 89, 98)( 90, 99)( 91,100)( 92,101)( 93,102)(109,163)(110,164)(111,165)(112,184)(113,185)(114,186)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,175)(122,176)(123,177)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,166)(131,167)(132,168)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,211)(140,212)(141,213)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,202)(149,203)(150,204)(151,196)(152,197)(153,198)(154,199)(155,200)(156,201)(157,193)(158,194)(159,195)(160,214)(161,215)(162,216);
s1 := Sym(216)!(  1,109)(  2,111)(  3,110)(  4,112)(  5,114)(  6,113)(  7,115)(  8,117)(  9,116)( 10,133)( 11,135)( 12,134)( 13,127)( 14,129)( 15,128)( 16,130)( 17,132)( 18,131)( 19,121)( 20,123)( 21,122)( 22,124)( 23,126)( 24,125)( 25,118)( 26,120)( 27,119)( 28,136)( 29,138)( 30,137)( 31,139)( 32,141)( 33,140)( 34,142)( 35,144)( 36,143)( 37,160)( 38,162)( 39,161)( 40,154)( 41,156)( 42,155)( 43,157)( 44,159)( 45,158)( 46,148)( 47,150)( 48,149)( 49,151)( 50,153)( 51,152)( 52,145)( 53,147)( 54,146)( 55,163)( 56,165)( 57,164)( 58,166)( 59,168)( 60,167)( 61,169)( 62,171)( 63,170)( 64,187)( 65,189)( 66,188)( 67,181)( 68,183)( 69,182)( 70,184)( 71,186)( 72,185)( 73,175)( 74,177)( 75,176)( 76,178)( 77,180)( 78,179)( 79,172)( 80,174)( 81,173)( 82,190)( 83,192)( 84,191)( 85,193)( 86,195)( 87,194)( 88,196)( 89,198)( 90,197)( 91,214)( 92,216)( 93,215)( 94,208)( 95,210)( 96,209)( 97,211)( 98,213)( 99,212)(100,202)(101,204)(102,203)(103,205)(104,207)(105,206)(106,199)(107,201)(108,200);
s2 := Sym(216)!(  1, 68)(  2, 67)(  3, 69)(  4, 65)(  5, 64)(  6, 66)(  7, 71)(  8, 70)(  9, 72)( 10, 59)( 11, 58)( 12, 60)( 13, 56)( 14, 55)( 15, 57)( 16, 62)( 17, 61)( 18, 63)( 19, 77)( 20, 76)( 21, 78)( 22, 74)( 23, 73)( 24, 75)( 25, 80)( 26, 79)( 27, 81)( 28, 95)( 29, 94)( 30, 96)( 31, 92)( 32, 91)( 33, 93)( 34, 98)( 35, 97)( 36, 99)( 37, 86)( 38, 85)( 39, 87)( 40, 83)( 41, 82)( 42, 84)( 43, 89)( 44, 88)( 45, 90)( 46,104)( 47,103)( 48,105)( 49,101)( 50,100)( 51,102)( 52,107)( 53,106)( 54,108)(109,203)(110,202)(111,204)(112,200)(113,199)(114,201)(115,206)(116,205)(117,207)(118,194)(119,193)(120,195)(121,191)(122,190)(123,192)(124,197)(125,196)(126,198)(127,212)(128,211)(129,213)(130,209)(131,208)(132,210)(133,215)(134,214)(135,216)(136,176)(137,175)(138,177)(139,173)(140,172)(141,174)(142,179)(143,178)(144,180)(145,167)(146,166)(147,168)(148,164)(149,163)(150,165)(151,170)(152,169)(153,171)(154,185)(155,184)(156,186)(157,182)(158,181)(159,183)(160,188)(161,187)(162,189);
poly := sub<Sym(216)|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*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*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 >;

References

None.

to this polytope.

Twisty Puzzle