Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,24}

Atlas Canonical Name {6,24}*1152g

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1152,155812)
Rank
3
Schläfli Type
{6,24}
Vertices, edges, …
24, 288, 96
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

16-fold

24-fold

32-fold

36-fold

48-fold

72-fold

96-fold

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

48 facets

12 vertex figures

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

48 facets

12 vertex figures

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

48 facets

8 vertex figures

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

24 facets

6 vertex figures

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

24 facets

6 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s0*s1*(s0*(s2*s1)^2)^2*s0*(s2*s1)^7*s2> of order 4

24 facets

6 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, s1*(s0*(s2*s1)^2)^2*s0*(s2*s1)^7*s2> of order 4

24 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143)(147,148)(151,152)(155,156)(157,169)(158,170)(159,172)(160,171)(161,173)(162,174)(163,176)(164,175)(165,177)(166,178)(167,180)(168,179)(183,184)(187,188)(191,192)(193,205)(194,206)(195,208)(196,207)(197,209)(198,210)(199,212)(200,211)(201,213)(202,214)(203,216)(204,215)(219,220)(223,224)(227,228)(229,241)(230,242)(231,244)(232,243)(233,245)(234,246)(235,248)(236,247)(237,249)(238,250)(239,252)(240,251)(255,256)(259,260)(263,264)(265,277)(266,278)(267,280)(268,279)(269,281)(270,282)(271,284)(272,283)(273,285)(274,286)(275,288)(276,287);;
s1 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)(  8, 22)(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 26, 28)( 29, 33)( 30, 36)( 31, 35)( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)( 66, 72)( 67, 71)( 68, 70)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)( 78,132)( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,141)(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)(108,138)(145,229)(146,232)(147,231)(148,230)(149,237)(150,240)(151,239)(152,238)(153,233)(154,236)(155,235)(156,234)(157,217)(158,220)(159,219)(160,218)(161,225)(162,228)(163,227)(164,226)(165,221)(166,224)(167,223)(168,222)(169,241)(170,244)(171,243)(172,242)(173,249)(174,252)(175,251)(176,250)(177,245)(178,248)(179,247)(180,246)(181,265)(182,268)(183,267)(184,266)(185,273)(186,276)(187,275)(188,274)(189,269)(190,272)(191,271)(192,270)(193,253)(194,256)(195,255)(196,254)(197,261)(198,264)(199,263)(200,262)(201,257)(202,260)(203,259)(204,258)(205,277)(206,280)(207,279)(208,278)(209,285)(210,288)(211,287)(212,286)(213,281)(214,284)(215,283)(216,282);;
s2 := (  1,150)(  2,149)(  3,151)(  4,152)(  5,146)(  6,145)(  7,147)(  8,148)(  9,154)( 10,153)( 11,155)( 12,156)( 13,174)( 14,173)( 15,175)( 16,176)( 17,170)( 18,169)( 19,171)( 20,172)( 21,178)( 22,177)( 23,179)( 24,180)( 25,162)( 26,161)( 27,163)( 28,164)( 29,158)( 30,157)( 31,159)( 32,160)( 33,166)( 34,165)( 35,167)( 36,168)( 37,186)( 38,185)( 39,187)( 40,188)( 41,182)( 42,181)( 43,183)( 44,184)( 45,190)( 46,189)( 47,191)( 48,192)( 49,210)( 50,209)( 51,211)( 52,212)( 53,206)( 54,205)( 55,207)( 56,208)( 57,214)( 58,213)( 59,215)( 60,216)( 61,198)( 62,197)( 63,199)( 64,200)( 65,194)( 66,193)( 67,195)( 68,196)( 69,202)( 70,201)( 71,203)( 72,204)( 73,258)( 74,257)( 75,259)( 76,260)( 77,254)( 78,253)( 79,255)( 80,256)( 81,262)( 82,261)( 83,263)( 84,264)( 85,282)( 86,281)( 87,283)( 88,284)( 89,278)( 90,277)( 91,279)( 92,280)( 93,286)( 94,285)( 95,287)( 96,288)( 97,270)( 98,269)( 99,271)(100,272)(101,266)(102,265)(103,267)(104,268)(105,274)(106,273)(107,275)(108,276)(109,222)(110,221)(111,223)(112,224)(113,218)(114,217)(115,219)(116,220)(117,226)(118,225)(119,227)(120,228)(121,246)(122,245)(123,247)(124,248)(125,242)(126,241)(127,243)(128,244)(129,250)(130,249)(131,251)(132,252)(133,234)(134,233)(135,235)(136,236)(137,230)(138,229)(139,231)(140,232)(141,238)(142,237)(143,239)(144,240);;
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*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*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*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(288)!(  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143)(147,148)(151,152)(155,156)(157,169)(158,170)(159,172)(160,171)(161,173)(162,174)(163,176)(164,175)(165,177)(166,178)(167,180)(168,179)(183,184)(187,188)(191,192)(193,205)(194,206)(195,208)(196,207)(197,209)(198,210)(199,212)(200,211)(201,213)(202,214)(203,216)(204,215)(219,220)(223,224)(227,228)(229,241)(230,242)(231,244)(232,243)(233,245)(234,246)(235,248)(236,247)(237,249)(238,250)(239,252)(240,251)(255,256)(259,260)(263,264)(265,277)(266,278)(267,280)(268,279)(269,281)(270,282)(271,284)(272,283)(273,285)(274,286)(275,288)(276,287);
s1 := Sym(288)!(  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 21)(  6, 24)(  7, 23)(  8, 22)(  9, 17)( 10, 20)( 11, 19)( 12, 18)( 26, 28)( 29, 33)( 30, 36)( 31, 35)( 32, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)( 62, 64)( 65, 69)( 66, 72)( 67, 71)( 68, 70)( 73,121)( 74,124)( 75,123)( 76,122)( 77,129)( 78,132)( 79,131)( 80,130)( 81,125)( 82,128)( 83,127)( 84,126)( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,141)(102,144)(103,143)(104,142)(105,137)(106,140)(107,139)(108,138)(145,229)(146,232)(147,231)(148,230)(149,237)(150,240)(151,239)(152,238)(153,233)(154,236)(155,235)(156,234)(157,217)(158,220)(159,219)(160,218)(161,225)(162,228)(163,227)(164,226)(165,221)(166,224)(167,223)(168,222)(169,241)(170,244)(171,243)(172,242)(173,249)(174,252)(175,251)(176,250)(177,245)(178,248)(179,247)(180,246)(181,265)(182,268)(183,267)(184,266)(185,273)(186,276)(187,275)(188,274)(189,269)(190,272)(191,271)(192,270)(193,253)(194,256)(195,255)(196,254)(197,261)(198,264)(199,263)(200,262)(201,257)(202,260)(203,259)(204,258)(205,277)(206,280)(207,279)(208,278)(209,285)(210,288)(211,287)(212,286)(213,281)(214,284)(215,283)(216,282);
s2 := Sym(288)!(  1,150)(  2,149)(  3,151)(  4,152)(  5,146)(  6,145)(  7,147)(  8,148)(  9,154)( 10,153)( 11,155)( 12,156)( 13,174)( 14,173)( 15,175)( 16,176)( 17,170)( 18,169)( 19,171)( 20,172)( 21,178)( 22,177)( 23,179)( 24,180)( 25,162)( 26,161)( 27,163)( 28,164)( 29,158)( 30,157)( 31,159)( 32,160)( 33,166)( 34,165)( 35,167)( 36,168)( 37,186)( 38,185)( 39,187)( 40,188)( 41,182)( 42,181)( 43,183)( 44,184)( 45,190)( 46,189)( 47,191)( 48,192)( 49,210)( 50,209)( 51,211)( 52,212)( 53,206)( 54,205)( 55,207)( 56,208)( 57,214)( 58,213)( 59,215)( 60,216)( 61,198)( 62,197)( 63,199)( 64,200)( 65,194)( 66,193)( 67,195)( 68,196)( 69,202)( 70,201)( 71,203)( 72,204)( 73,258)( 74,257)( 75,259)( 76,260)( 77,254)( 78,253)( 79,255)( 80,256)( 81,262)( 82,261)( 83,263)( 84,264)( 85,282)( 86,281)( 87,283)( 88,284)( 89,278)( 90,277)( 91,279)( 92,280)( 93,286)( 94,285)( 95,287)( 96,288)( 97,270)( 98,269)( 99,271)(100,272)(101,266)(102,265)(103,267)(104,268)(105,274)(106,273)(107,275)(108,276)(109,222)(110,221)(111,223)(112,224)(113,218)(114,217)(115,219)(116,220)(117,226)(118,225)(119,227)(120,228)(121,246)(122,245)(123,247)(124,248)(125,242)(126,241)(127,243)(128,244)(129,250)(130,249)(131,251)(132,252)(133,234)(134,233)(135,235)(136,236)(137,230)(138,229)(139,231)(140,232)(141,238)(142,237)(143,239)(144,240);
poly := sub<Sym(288)|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*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*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*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