Part of the Atlas of Small Regular Polytopes

Polytope of Type {24,4}

Atlas Canonical Name {24,4}*768f

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(768,1085644)
Rank
3
Schläfli Type
{24,4}
Vertices, edges, …
96, 192, 16
Order of s0s1s2
24
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

32-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=<s1*s0*s1*s2*s1*s0*s2*s1> of order 2

8 facets

56 vertex figures

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

8 facets

48 vertex figures

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

8 facets

48 vertex figures

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

4 facets

32 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 4

4 facets

36 vertex figures

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

4 facets

24 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)(100,147)(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)(108,159)(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)(116,179)(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)(124,191)(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)(132,163)(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)(143,172)(144,171)(193,289)(194,290)(195,292)(196,291)(197,293)(198,294)(199,296)(200,295)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)(208,299)(209,321)(210,322)(211,324)(212,323)(213,325)(214,326)(215,328)(216,327)(217,333)(218,334)(219,336)(220,335)(221,329)(222,330)(223,332)(224,331)(225,305)(226,306)(227,308)(228,307)(229,309)(230,310)(231,312)(232,311)(233,317)(234,318)(235,320)(236,319)(237,313)(238,314)(239,316)(240,315)(241,337)(242,338)(243,340)(244,339)(245,341)(246,342)(247,344)(248,343)(249,349)(250,350)(251,352)(252,351)(253,345)(254,346)(255,348)(256,347)(257,369)(258,370)(259,372)(260,371)(261,373)(262,374)(263,376)(264,375)(265,381)(266,382)(267,384)(268,383)(269,377)(270,378)(271,380)(272,379)(273,353)(274,354)(275,356)(276,355)(277,357)(278,358)(279,360)(280,359)(281,365)(282,366)(283,368)(284,367)(285,361)(286,362)(287,364)(288,363);;
s1 := (  1,321)(  2,323)(  3,322)(  4,324)(  5,331)(  6,329)(  7,332)(  8,330)(  9,326)( 10,328)( 11,325)( 12,327)( 13,336)( 14,334)( 15,335)( 16,333)( 17,305)( 18,307)( 19,306)( 20,308)( 21,315)( 22,313)( 23,316)( 24,314)( 25,310)( 26,312)( 27,309)( 28,311)( 29,320)( 30,318)( 31,319)( 32,317)( 33,289)( 34,291)( 35,290)( 36,292)( 37,299)( 38,297)( 39,300)( 40,298)( 41,294)( 42,296)( 43,293)( 44,295)( 45,304)( 46,302)( 47,303)( 48,301)( 49,369)( 50,371)( 51,370)( 52,372)( 53,379)( 54,377)( 55,380)( 56,378)( 57,374)( 58,376)( 59,373)( 60,375)( 61,384)( 62,382)( 63,383)( 64,381)( 65,353)( 66,355)( 67,354)( 68,356)( 69,363)( 70,361)( 71,364)( 72,362)( 73,358)( 74,360)( 75,357)( 76,359)( 77,368)( 78,366)( 79,367)( 80,365)( 81,337)( 82,339)( 83,338)( 84,340)( 85,347)( 86,345)( 87,348)( 88,346)( 89,342)( 90,344)( 91,341)( 92,343)( 93,352)( 94,350)( 95,351)( 96,349)( 97,225)( 98,227)( 99,226)(100,228)(101,235)(102,233)(103,236)(104,234)(105,230)(106,232)(107,229)(108,231)(109,240)(110,238)(111,239)(112,237)(113,209)(114,211)(115,210)(116,212)(117,219)(118,217)(119,220)(120,218)(121,214)(122,216)(123,213)(124,215)(125,224)(126,222)(127,223)(128,221)(129,193)(130,195)(131,194)(132,196)(133,203)(134,201)(135,204)(136,202)(137,198)(138,200)(139,197)(140,199)(141,208)(142,206)(143,207)(144,205)(145,273)(146,275)(147,274)(148,276)(149,283)(150,281)(151,284)(152,282)(153,278)(154,280)(155,277)(156,279)(157,288)(158,286)(159,287)(160,285)(161,257)(162,259)(163,258)(164,260)(165,267)(166,265)(167,268)(168,266)(169,262)(170,264)(171,261)(172,263)(173,272)(174,270)(175,271)(176,269)(177,241)(178,243)(179,242)(180,244)(181,251)(182,249)(183,252)(184,250)(185,246)(186,248)(187,245)(188,247)(189,256)(190,254)(191,255)(192,253);;
s2 := (  1, 53)(  2, 54)(  3, 55)(  4, 56)(  5, 49)(  6, 50)(  7, 51)(  8, 52)(  9, 61)( 10, 62)( 11, 63)( 12, 64)( 13, 57)( 14, 58)( 15, 59)( 16, 60)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 65)( 22, 66)( 23, 67)( 24, 68)( 25, 77)( 26, 78)( 27, 79)( 28, 80)( 29, 73)( 30, 74)( 31, 75)( 32, 76)( 33, 85)( 34, 86)( 35, 87)( 36, 88)( 37, 81)( 38, 82)( 39, 83)( 40, 84)( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 89)( 46, 90)( 47, 91)( 48, 92)( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188)(193,245)(194,246)(195,247)(196,248)(197,241)(198,242)(199,243)(200,244)(201,253)(202,254)(203,255)(204,256)(205,249)(206,250)(207,251)(208,252)(209,261)(210,262)(211,263)(212,264)(213,257)(214,258)(215,259)(216,260)(217,269)(218,270)(219,271)(220,272)(221,265)(222,266)(223,267)(224,268)(225,277)(226,278)(227,279)(228,280)(229,273)(230,274)(231,275)(232,276)(233,285)(234,286)(235,287)(236,288)(237,281)(238,282)(239,283)(240,284)(289,341)(290,342)(291,343)(292,344)(293,337)(294,338)(295,339)(296,340)(297,349)(298,350)(299,351)(300,352)(301,345)(302,346)(303,347)(304,348)(305,357)(306,358)(307,359)(308,360)(309,353)(310,354)(311,355)(312,356)(313,365)(314,366)(315,367)(316,368)(317,361)(318,362)(319,363)(320,364)(321,373)(322,374)(323,375)(324,376)(325,369)(326,370)(327,371)(328,372)(329,381)(330,382)(331,383)(332,384)(333,377)(334,378)(335,379)(336,380);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(384)!(  3,  4)(  7,  8)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)(100,147)(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)(108,159)(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)(116,179)(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)(124,191)(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)(132,163)(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)(140,175)(141,169)(142,170)(143,172)(144,171)(193,289)(194,290)(195,292)(196,291)(197,293)(198,294)(199,296)(200,295)(201,301)(202,302)(203,304)(204,303)(205,297)(206,298)(207,300)(208,299)(209,321)(210,322)(211,324)(212,323)(213,325)(214,326)(215,328)(216,327)(217,333)(218,334)(219,336)(220,335)(221,329)(222,330)(223,332)(224,331)(225,305)(226,306)(227,308)(228,307)(229,309)(230,310)(231,312)(232,311)(233,317)(234,318)(235,320)(236,319)(237,313)(238,314)(239,316)(240,315)(241,337)(242,338)(243,340)(244,339)(245,341)(246,342)(247,344)(248,343)(249,349)(250,350)(251,352)(252,351)(253,345)(254,346)(255,348)(256,347)(257,369)(258,370)(259,372)(260,371)(261,373)(262,374)(263,376)(264,375)(265,381)(266,382)(267,384)(268,383)(269,377)(270,378)(271,380)(272,379)(273,353)(274,354)(275,356)(276,355)(277,357)(278,358)(279,360)(280,359)(281,365)(282,366)(283,368)(284,367)(285,361)(286,362)(287,364)(288,363);
s1 := Sym(384)!(  1,321)(  2,323)(  3,322)(  4,324)(  5,331)(  6,329)(  7,332)(  8,330)(  9,326)( 10,328)( 11,325)( 12,327)( 13,336)( 14,334)( 15,335)( 16,333)( 17,305)( 18,307)( 19,306)( 20,308)( 21,315)( 22,313)( 23,316)( 24,314)( 25,310)( 26,312)( 27,309)( 28,311)( 29,320)( 30,318)( 31,319)( 32,317)( 33,289)( 34,291)( 35,290)( 36,292)( 37,299)( 38,297)( 39,300)( 40,298)( 41,294)( 42,296)( 43,293)( 44,295)( 45,304)( 46,302)( 47,303)( 48,301)( 49,369)( 50,371)( 51,370)( 52,372)( 53,379)( 54,377)( 55,380)( 56,378)( 57,374)( 58,376)( 59,373)( 60,375)( 61,384)( 62,382)( 63,383)( 64,381)( 65,353)( 66,355)( 67,354)( 68,356)( 69,363)( 70,361)( 71,364)( 72,362)( 73,358)( 74,360)( 75,357)( 76,359)( 77,368)( 78,366)( 79,367)( 80,365)( 81,337)( 82,339)( 83,338)( 84,340)( 85,347)( 86,345)( 87,348)( 88,346)( 89,342)( 90,344)( 91,341)( 92,343)( 93,352)( 94,350)( 95,351)( 96,349)( 97,225)( 98,227)( 99,226)(100,228)(101,235)(102,233)(103,236)(104,234)(105,230)(106,232)(107,229)(108,231)(109,240)(110,238)(111,239)(112,237)(113,209)(114,211)(115,210)(116,212)(117,219)(118,217)(119,220)(120,218)(121,214)(122,216)(123,213)(124,215)(125,224)(126,222)(127,223)(128,221)(129,193)(130,195)(131,194)(132,196)(133,203)(134,201)(135,204)(136,202)(137,198)(138,200)(139,197)(140,199)(141,208)(142,206)(143,207)(144,205)(145,273)(146,275)(147,274)(148,276)(149,283)(150,281)(151,284)(152,282)(153,278)(154,280)(155,277)(156,279)(157,288)(158,286)(159,287)(160,285)(161,257)(162,259)(163,258)(164,260)(165,267)(166,265)(167,268)(168,266)(169,262)(170,264)(171,261)(172,263)(173,272)(174,270)(175,271)(176,269)(177,241)(178,243)(179,242)(180,244)(181,251)(182,249)(183,252)(184,250)(185,246)(186,248)(187,245)(188,247)(189,256)(190,254)(191,255)(192,253);
s2 := Sym(384)!(  1, 53)(  2, 54)(  3, 55)(  4, 56)(  5, 49)(  6, 50)(  7, 51)(  8, 52)(  9, 61)( 10, 62)( 11, 63)( 12, 64)( 13, 57)( 14, 58)( 15, 59)( 16, 60)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 65)( 22, 66)( 23, 67)( 24, 68)( 25, 77)( 26, 78)( 27, 79)( 28, 80)( 29, 73)( 30, 74)( 31, 75)( 32, 76)( 33, 85)( 34, 86)( 35, 87)( 36, 88)( 37, 81)( 38, 82)( 39, 83)( 40, 84)( 41, 93)( 42, 94)( 43, 95)( 44, 96)( 45, 89)( 46, 90)( 47, 91)( 48, 92)( 97,149)( 98,150)( 99,151)(100,152)(101,145)(102,146)(103,147)(104,148)(105,157)(106,158)(107,159)(108,160)(109,153)(110,154)(111,155)(112,156)(113,165)(114,166)(115,167)(116,168)(117,161)(118,162)(119,163)(120,164)(121,173)(122,174)(123,175)(124,176)(125,169)(126,170)(127,171)(128,172)(129,181)(130,182)(131,183)(132,184)(133,177)(134,178)(135,179)(136,180)(137,189)(138,190)(139,191)(140,192)(141,185)(142,186)(143,187)(144,188)(193,245)(194,246)(195,247)(196,248)(197,241)(198,242)(199,243)(200,244)(201,253)(202,254)(203,255)(204,256)(205,249)(206,250)(207,251)(208,252)(209,261)(210,262)(211,263)(212,264)(213,257)(214,258)(215,259)(216,260)(217,269)(218,270)(219,271)(220,272)(221,265)(222,266)(223,267)(224,268)(225,277)(226,278)(227,279)(228,280)(229,273)(230,274)(231,275)(232,276)(233,285)(234,286)(235,287)(236,288)(237,281)(238,282)(239,283)(240,284)(289,341)(290,342)(291,343)(292,344)(293,337)(294,338)(295,339)(296,340)(297,349)(298,350)(299,351)(300,352)(301,345)(302,346)(303,347)(304,348)(305,357)(306,358)(307,359)(308,360)(309,353)(310,354)(311,355)(312,356)(313,365)(314,366)(315,367)(316,368)(317,361)(318,362)(319,363)(320,364)(321,373)(322,374)(323,375)(324,376)(325,369)(326,370)(327,371)(328,372)(329,381)(330,382)(331,383)(332,384)(333,377)(334,378)(335,379)(336,380);
poly := sub<Sym(384)|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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle