Polytope of Type {4,24,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,24,4}*768n
if this polytope has a name.
Group : SmallGroup(768,1090111)
Rank : 4
Schlafli Type : {4,24,4}
Number of vertices, edges, etc : 4, 48, 48, 4
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,12,4}*384f
4-fold quotients : {4,6,4}*192d
8-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239)
(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)(254,255)
(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)(270,271)
(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)(286,287)
(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)(302,303)
(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)(318,319)
(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)(334,335)
(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)(350,351)
(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)(366,367)
(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)(382,383);;
s1 := ( 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);;
s2 := ( 1,321)( 2,323)( 3,322)( 4,324)( 5,329)( 6,331)( 7,330)( 8,332)
( 9,325)( 10,327)( 11,326)( 12,328)( 13,333)( 14,335)( 15,334)( 16,336)
( 17,305)( 18,307)( 19,306)( 20,308)( 21,313)( 22,315)( 23,314)( 24,316)
( 25,309)( 26,311)( 27,310)( 28,312)( 29,317)( 30,319)( 31,318)( 32,320)
( 33,289)( 34,291)( 35,290)( 36,292)( 37,297)( 38,299)( 39,298)( 40,300)
( 41,293)( 42,295)( 43,294)( 44,296)( 45,301)( 46,303)( 47,302)( 48,304)
( 49,369)( 50,371)( 51,370)( 52,372)( 53,377)( 54,379)( 55,378)( 56,380)
( 57,373)( 58,375)( 59,374)( 60,376)( 61,381)( 62,383)( 63,382)( 64,384)
( 65,353)( 66,355)( 67,354)( 68,356)( 69,361)( 70,363)( 71,362)( 72,364)
( 73,357)( 74,359)( 75,358)( 76,360)( 77,365)( 78,367)( 79,366)( 80,368)
( 81,337)( 82,339)( 83,338)( 84,340)( 85,345)( 86,347)( 87,346)( 88,348)
( 89,341)( 90,343)( 91,342)( 92,344)( 93,349)( 94,351)( 95,350)( 96,352)
( 97,225)( 98,227)( 99,226)(100,228)(101,233)(102,235)(103,234)(104,236)
(105,229)(106,231)(107,230)(108,232)(109,237)(110,239)(111,238)(112,240)
(113,209)(114,211)(115,210)(116,212)(117,217)(118,219)(119,218)(120,220)
(121,213)(122,215)(123,214)(124,216)(125,221)(126,223)(127,222)(128,224)
(129,193)(130,195)(131,194)(132,196)(133,201)(134,203)(135,202)(136,204)
(137,197)(138,199)(139,198)(140,200)(141,205)(142,207)(143,206)(144,208)
(145,273)(146,275)(147,274)(148,276)(149,281)(150,283)(151,282)(152,284)
(153,277)(154,279)(155,278)(156,280)(157,285)(158,287)(159,286)(160,288)
(161,257)(162,259)(163,258)(164,260)(165,265)(166,267)(167,266)(168,268)
(169,261)(170,263)(171,262)(172,264)(173,269)(174,271)(175,270)(176,272)
(177,241)(178,243)(179,242)(180,244)(181,249)(182,251)(183,250)(184,252)
(185,245)(186,247)(187,246)(188,248)(189,253)(190,255)(191,254)(192,256);;
s3 := ( 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,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*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(384)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239)(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)
(254,255)(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)
(270,271)(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)
(286,287)(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)
(302,303)(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)
(318,319)(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)
(334,335)(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)
(350,351)(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)
(366,367)(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)
(382,383);
s1 := 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);
s2 := Sym(384)!( 1,321)( 2,323)( 3,322)( 4,324)( 5,329)( 6,331)( 7,330)
( 8,332)( 9,325)( 10,327)( 11,326)( 12,328)( 13,333)( 14,335)( 15,334)
( 16,336)( 17,305)( 18,307)( 19,306)( 20,308)( 21,313)( 22,315)( 23,314)
( 24,316)( 25,309)( 26,311)( 27,310)( 28,312)( 29,317)( 30,319)( 31,318)
( 32,320)( 33,289)( 34,291)( 35,290)( 36,292)( 37,297)( 38,299)( 39,298)
( 40,300)( 41,293)( 42,295)( 43,294)( 44,296)( 45,301)( 46,303)( 47,302)
( 48,304)( 49,369)( 50,371)( 51,370)( 52,372)( 53,377)( 54,379)( 55,378)
( 56,380)( 57,373)( 58,375)( 59,374)( 60,376)( 61,381)( 62,383)( 63,382)
( 64,384)( 65,353)( 66,355)( 67,354)( 68,356)( 69,361)( 70,363)( 71,362)
( 72,364)( 73,357)( 74,359)( 75,358)( 76,360)( 77,365)( 78,367)( 79,366)
( 80,368)( 81,337)( 82,339)( 83,338)( 84,340)( 85,345)( 86,347)( 87,346)
( 88,348)( 89,341)( 90,343)( 91,342)( 92,344)( 93,349)( 94,351)( 95,350)
( 96,352)( 97,225)( 98,227)( 99,226)(100,228)(101,233)(102,235)(103,234)
(104,236)(105,229)(106,231)(107,230)(108,232)(109,237)(110,239)(111,238)
(112,240)(113,209)(114,211)(115,210)(116,212)(117,217)(118,219)(119,218)
(120,220)(121,213)(122,215)(123,214)(124,216)(125,221)(126,223)(127,222)
(128,224)(129,193)(130,195)(131,194)(132,196)(133,201)(134,203)(135,202)
(136,204)(137,197)(138,199)(139,198)(140,200)(141,205)(142,207)(143,206)
(144,208)(145,273)(146,275)(147,274)(148,276)(149,281)(150,283)(151,282)
(152,284)(153,277)(154,279)(155,278)(156,280)(157,285)(158,287)(159,286)
(160,288)(161,257)(162,259)(163,258)(164,260)(165,265)(166,267)(167,266)
(168,268)(169,261)(170,263)(171,262)(172,264)(173,269)(174,271)(175,270)
(176,272)(177,241)(178,243)(179,242)(180,244)(181,249)(182,251)(183,250)
(184,252)(185,245)(186,247)(187,246)(188,248)(189,253)(190,255)(191,254)
(192,256);
s3 := 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,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope