Part of the Atlas of Small Regular Polytopes

Polytope of Type {24,4}

Atlas Canonical Name {24,4}*1728b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,12703)
Rank
3
Schläfli Type
{24,4}
Vertices, edges, …
216, 432, 36
Order of s0s1s2
24
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

4-fold

6-fold

8-fold

12-fold

24-fold

27-fold

54-fold

108-fold

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

18 facets

108 vertex figures

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

18 facets

108 vertex figures

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

18 facets

120 vertex figures

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

12 facets

72 vertex figures

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

12 facets

72 vertex figures

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

6 facets

48 vertex figures

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

6 facets

48 vertex figures

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

6 facets

36 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s1*s0*s1*s2)^3> of order 6

6 facets

36 vertex figures

P/N, where N=<(s0*s1*s2*s1)^3, s1*s2*(s1*s0)^2*s1*s2*s1*s0*s1*s2> of order 6

6 facets

36 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle