Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*864b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(864,4000)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
36, 216, 72
Order of s0s1s2
6
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

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

18-fold

36-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=<s0*s1*s0*s2*s1*s0*(s1*s2)^4*s1> of order 2

36 facets

18 vertex figures

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

36 facets

24 vertex figures

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

36 facets

18 vertex figures

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

24 facets

20 vertex figures

P/N, where N=<(s1*s2)^6, (s0*s1)^2*(s2*s1)^4*s0*s1*s2> of order 4

18 facets

12 vertex figures

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

12 facets

16 vertex figures

P/N, where N=<s0*(s1*s2)^3*s1*s0*s1*s2*s1> of order 6

12 facets

12 vertex figures

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

12 facets

10 vertex figures

Representations

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