Polytope of Type {2,24,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,24,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30374)
Rank : 4
Schlafli Type : {2,24,6}
Number of vertices, edges, etc : 2, 72, 216, 18
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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 : {2,12,6}*864e
3-fold quotients : {2,8,6}*576
4-fold quotients : {2,12,6}*432b
6-fold quotients : {2,4,6}*288
12-fold quotients : {2,4,6}*144
27-fold quotients : {2,8,2}*64
54-fold quotients : {2,4,2}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 9, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 25)( 16, 24)
( 17, 26)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 33, 34)( 36, 38)( 39, 48)
( 40, 50)( 41, 49)( 42, 52)( 43, 51)( 44, 53)( 45, 56)( 46, 55)( 47, 54)
( 57, 84)( 58, 86)( 59, 85)( 60, 88)( 61, 87)( 62, 89)( 63, 92)( 64, 91)
( 65, 90)( 66,102)( 67,104)( 68,103)( 69,106)( 70,105)( 71,107)( 72,110)
( 73,109)( 74,108)( 75, 93)( 76, 95)( 77, 94)( 78, 97)( 79, 96)( 80, 98)
( 81,101)( 82,100)( 83, 99)(111,138)(112,140)(113,139)(114,142)(115,141)
(116,143)(117,146)(118,145)(119,144)(120,156)(121,158)(122,157)(123,160)
(124,159)(125,161)(126,164)(127,163)(128,162)(129,147)(130,149)(131,148)
(132,151)(133,150)(134,152)(135,155)(136,154)(137,153)(166,167)(168,169)
(171,173)(174,183)(175,185)(176,184)(177,187)(178,186)(179,188)(180,191)
(181,190)(182,189)(193,194)(195,196)(198,200)(201,210)(202,212)(203,211)
(204,214)(205,213)(206,215)(207,218)(208,217)(209,216)(219,273)(220,275)
(221,274)(222,277)(223,276)(224,278)(225,281)(226,280)(227,279)(228,291)
(229,293)(230,292)(231,295)(232,294)(233,296)(234,299)(235,298)(236,297)
(237,282)(238,284)(239,283)(240,286)(241,285)(242,287)(243,290)(244,289)
(245,288)(246,300)(247,302)(248,301)(249,304)(250,303)(251,305)(252,308)
(253,307)(254,306)(255,318)(256,320)(257,319)(258,322)(259,321)(260,323)
(261,326)(262,325)(263,324)(264,309)(265,311)(266,310)(267,313)(268,312)
(269,314)(270,317)(271,316)(272,315)(327,408)(328,410)(329,409)(330,412)
(331,411)(332,413)(333,416)(334,415)(335,414)(336,426)(337,428)(338,427)
(339,430)(340,429)(341,431)(342,434)(343,433)(344,432)(345,417)(346,419)
(347,418)(348,421)(349,420)(350,422)(351,425)(352,424)(353,423)(354,381)
(355,383)(356,382)(357,385)(358,384)(359,386)(360,389)(361,388)(362,387)
(363,399)(364,401)(365,400)(366,403)(367,402)(368,404)(369,407)(370,406)
(371,405)(372,390)(373,392)(374,391)(375,394)(376,393)(377,395)(378,398)
(379,397)(380,396);;
s2 := ( 3,220)( 4,219)( 5,221)( 6,228)( 7,230)( 8,229)( 9,239)( 10,238)
( 11,237)( 12,222)( 13,224)( 14,223)( 15,231)( 16,233)( 17,232)( 18,240)
( 19,242)( 20,241)( 21,227)( 22,226)( 23,225)( 24,234)( 25,236)( 26,235)
( 27,244)( 28,243)( 29,245)( 30,247)( 31,246)( 32,248)( 33,255)( 34,257)
( 35,256)( 36,266)( 37,265)( 38,264)( 39,249)( 40,251)( 41,250)( 42,258)
( 43,260)( 44,259)( 45,267)( 46,269)( 47,268)( 48,254)( 49,253)( 50,252)
( 51,261)( 52,263)( 53,262)( 54,271)( 55,270)( 56,272)( 57,301)( 58,300)
( 59,302)( 60,309)( 61,311)( 62,310)( 63,320)( 64,319)( 65,318)( 66,303)
( 67,305)( 68,304)( 69,312)( 70,314)( 71,313)( 72,321)( 73,323)( 74,322)
( 75,308)( 76,307)( 77,306)( 78,315)( 79,317)( 80,316)( 81,325)( 82,324)
( 83,326)( 84,274)( 85,273)( 86,275)( 87,282)( 88,284)( 89,283)( 90,293)
( 91,292)( 92,291)( 93,276)( 94,278)( 95,277)( 96,285)( 97,287)( 98,286)
( 99,294)(100,296)(101,295)(102,281)(103,280)(104,279)(105,288)(106,290)
(107,289)(108,298)(109,297)(110,299)(111,355)(112,354)(113,356)(114,363)
(115,365)(116,364)(117,374)(118,373)(119,372)(120,357)(121,359)(122,358)
(123,366)(124,368)(125,367)(126,375)(127,377)(128,376)(129,362)(130,361)
(131,360)(132,369)(133,371)(134,370)(135,379)(136,378)(137,380)(138,328)
(139,327)(140,329)(141,336)(142,338)(143,337)(144,347)(145,346)(146,345)
(147,330)(148,332)(149,331)(150,339)(151,341)(152,340)(153,348)(154,350)
(155,349)(156,335)(157,334)(158,333)(159,342)(160,344)(161,343)(162,352)
(163,351)(164,353)(165,382)(166,381)(167,383)(168,390)(169,392)(170,391)
(171,401)(172,400)(173,399)(174,384)(175,386)(176,385)(177,393)(178,395)
(179,394)(180,402)(181,404)(182,403)(183,389)(184,388)(185,387)(186,396)
(187,398)(188,397)(189,406)(190,405)(191,407)(192,409)(193,408)(194,410)
(195,417)(196,419)(197,418)(198,428)(199,427)(200,426)(201,411)(202,413)
(203,412)(204,420)(205,422)(206,421)(207,429)(208,431)(209,430)(210,416)
(211,415)(212,414)(213,423)(214,425)(215,424)(216,433)(217,432)(218,434);;
s3 := ( 3,172)( 4,173)( 5,171)( 6,168)( 7,169)( 8,170)( 9,167)( 10,165)
( 11,166)( 12,190)( 13,191)( 14,189)( 15,186)( 16,187)( 17,188)( 18,185)
( 19,183)( 20,184)( 21,181)( 22,182)( 23,180)( 24,177)( 25,178)( 26,179)
( 27,176)( 28,174)( 29,175)( 30,199)( 31,200)( 32,198)( 33,195)( 34,196)
( 35,197)( 36,194)( 37,192)( 38,193)( 39,217)( 40,218)( 41,216)( 42,213)
( 43,214)( 44,215)( 45,212)( 46,210)( 47,211)( 48,208)( 49,209)( 50,207)
( 51,204)( 52,205)( 53,206)( 54,203)( 55,201)( 56,202)( 57,145)( 58,146)
( 59,144)( 60,141)( 61,142)( 62,143)( 63,140)( 64,138)( 65,139)( 66,163)
( 67,164)( 68,162)( 69,159)( 70,160)( 71,161)( 72,158)( 73,156)( 74,157)
( 75,154)( 76,155)( 77,153)( 78,150)( 79,151)( 80,152)( 81,149)( 82,147)
( 83,148)( 84,118)( 85,119)( 86,117)( 87,114)( 88,115)( 89,116)( 90,113)
( 91,111)( 92,112)( 93,136)( 94,137)( 95,135)( 96,132)( 97,133)( 98,134)
( 99,131)(100,129)(101,130)(102,127)(103,128)(104,126)(105,123)(106,124)
(107,125)(108,122)(109,120)(110,121)(219,388)(220,389)(221,387)(222,384)
(223,385)(224,386)(225,383)(226,381)(227,382)(228,406)(229,407)(230,405)
(231,402)(232,403)(233,404)(234,401)(235,399)(236,400)(237,397)(238,398)
(239,396)(240,393)(241,394)(242,395)(243,392)(244,390)(245,391)(246,415)
(247,416)(248,414)(249,411)(250,412)(251,413)(252,410)(253,408)(254,409)
(255,433)(256,434)(257,432)(258,429)(259,430)(260,431)(261,428)(262,426)
(263,427)(264,424)(265,425)(266,423)(267,420)(268,421)(269,422)(270,419)
(271,417)(272,418)(273,361)(274,362)(275,360)(276,357)(277,358)(278,359)
(279,356)(280,354)(281,355)(282,379)(283,380)(284,378)(285,375)(286,376)
(287,377)(288,374)(289,372)(290,373)(291,370)(292,371)(293,369)(294,366)
(295,367)(296,368)(297,365)(298,363)(299,364)(300,334)(301,335)(302,333)
(303,330)(304,331)(305,332)(306,329)(307,327)(308,328)(309,352)(310,353)
(311,351)(312,348)(313,349)(314,350)(315,347)(316,345)(317,346)(318,343)
(319,344)(320,342)(321,339)(322,340)(323,341)(324,338)(325,336)(326,337);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(434)!(1,2);
s1 := Sym(434)!( 4, 5)( 6, 7)( 9, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 25)
( 16, 24)( 17, 26)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 33, 34)( 36, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 52)( 43, 51)( 44, 53)( 45, 56)( 46, 55)
( 47, 54)( 57, 84)( 58, 86)( 59, 85)( 60, 88)( 61, 87)( 62, 89)( 63, 92)
( 64, 91)( 65, 90)( 66,102)( 67,104)( 68,103)( 69,106)( 70,105)( 71,107)
( 72,110)( 73,109)( 74,108)( 75, 93)( 76, 95)( 77, 94)( 78, 97)( 79, 96)
( 80, 98)( 81,101)( 82,100)( 83, 99)(111,138)(112,140)(113,139)(114,142)
(115,141)(116,143)(117,146)(118,145)(119,144)(120,156)(121,158)(122,157)
(123,160)(124,159)(125,161)(126,164)(127,163)(128,162)(129,147)(130,149)
(131,148)(132,151)(133,150)(134,152)(135,155)(136,154)(137,153)(166,167)
(168,169)(171,173)(174,183)(175,185)(176,184)(177,187)(178,186)(179,188)
(180,191)(181,190)(182,189)(193,194)(195,196)(198,200)(201,210)(202,212)
(203,211)(204,214)(205,213)(206,215)(207,218)(208,217)(209,216)(219,273)
(220,275)(221,274)(222,277)(223,276)(224,278)(225,281)(226,280)(227,279)
(228,291)(229,293)(230,292)(231,295)(232,294)(233,296)(234,299)(235,298)
(236,297)(237,282)(238,284)(239,283)(240,286)(241,285)(242,287)(243,290)
(244,289)(245,288)(246,300)(247,302)(248,301)(249,304)(250,303)(251,305)
(252,308)(253,307)(254,306)(255,318)(256,320)(257,319)(258,322)(259,321)
(260,323)(261,326)(262,325)(263,324)(264,309)(265,311)(266,310)(267,313)
(268,312)(269,314)(270,317)(271,316)(272,315)(327,408)(328,410)(329,409)
(330,412)(331,411)(332,413)(333,416)(334,415)(335,414)(336,426)(337,428)
(338,427)(339,430)(340,429)(341,431)(342,434)(343,433)(344,432)(345,417)
(346,419)(347,418)(348,421)(349,420)(350,422)(351,425)(352,424)(353,423)
(354,381)(355,383)(356,382)(357,385)(358,384)(359,386)(360,389)(361,388)
(362,387)(363,399)(364,401)(365,400)(366,403)(367,402)(368,404)(369,407)
(370,406)(371,405)(372,390)(373,392)(374,391)(375,394)(376,393)(377,395)
(378,398)(379,397)(380,396);
s2 := Sym(434)!( 3,220)( 4,219)( 5,221)( 6,228)( 7,230)( 8,229)( 9,239)
( 10,238)( 11,237)( 12,222)( 13,224)( 14,223)( 15,231)( 16,233)( 17,232)
( 18,240)( 19,242)( 20,241)( 21,227)( 22,226)( 23,225)( 24,234)( 25,236)
( 26,235)( 27,244)( 28,243)( 29,245)( 30,247)( 31,246)( 32,248)( 33,255)
( 34,257)( 35,256)( 36,266)( 37,265)( 38,264)( 39,249)( 40,251)( 41,250)
( 42,258)( 43,260)( 44,259)( 45,267)( 46,269)( 47,268)( 48,254)( 49,253)
( 50,252)( 51,261)( 52,263)( 53,262)( 54,271)( 55,270)( 56,272)( 57,301)
( 58,300)( 59,302)( 60,309)( 61,311)( 62,310)( 63,320)( 64,319)( 65,318)
( 66,303)( 67,305)( 68,304)( 69,312)( 70,314)( 71,313)( 72,321)( 73,323)
( 74,322)( 75,308)( 76,307)( 77,306)( 78,315)( 79,317)( 80,316)( 81,325)
( 82,324)( 83,326)( 84,274)( 85,273)( 86,275)( 87,282)( 88,284)( 89,283)
( 90,293)( 91,292)( 92,291)( 93,276)( 94,278)( 95,277)( 96,285)( 97,287)
( 98,286)( 99,294)(100,296)(101,295)(102,281)(103,280)(104,279)(105,288)
(106,290)(107,289)(108,298)(109,297)(110,299)(111,355)(112,354)(113,356)
(114,363)(115,365)(116,364)(117,374)(118,373)(119,372)(120,357)(121,359)
(122,358)(123,366)(124,368)(125,367)(126,375)(127,377)(128,376)(129,362)
(130,361)(131,360)(132,369)(133,371)(134,370)(135,379)(136,378)(137,380)
(138,328)(139,327)(140,329)(141,336)(142,338)(143,337)(144,347)(145,346)
(146,345)(147,330)(148,332)(149,331)(150,339)(151,341)(152,340)(153,348)
(154,350)(155,349)(156,335)(157,334)(158,333)(159,342)(160,344)(161,343)
(162,352)(163,351)(164,353)(165,382)(166,381)(167,383)(168,390)(169,392)
(170,391)(171,401)(172,400)(173,399)(174,384)(175,386)(176,385)(177,393)
(178,395)(179,394)(180,402)(181,404)(182,403)(183,389)(184,388)(185,387)
(186,396)(187,398)(188,397)(189,406)(190,405)(191,407)(192,409)(193,408)
(194,410)(195,417)(196,419)(197,418)(198,428)(199,427)(200,426)(201,411)
(202,413)(203,412)(204,420)(205,422)(206,421)(207,429)(208,431)(209,430)
(210,416)(211,415)(212,414)(213,423)(214,425)(215,424)(216,433)(217,432)
(218,434);
s3 := Sym(434)!( 3,172)( 4,173)( 5,171)( 6,168)( 7,169)( 8,170)( 9,167)
( 10,165)( 11,166)( 12,190)( 13,191)( 14,189)( 15,186)( 16,187)( 17,188)
( 18,185)( 19,183)( 20,184)( 21,181)( 22,182)( 23,180)( 24,177)( 25,178)
( 26,179)( 27,176)( 28,174)( 29,175)( 30,199)( 31,200)( 32,198)( 33,195)
( 34,196)( 35,197)( 36,194)( 37,192)( 38,193)( 39,217)( 40,218)( 41,216)
( 42,213)( 43,214)( 44,215)( 45,212)( 46,210)( 47,211)( 48,208)( 49,209)
( 50,207)( 51,204)( 52,205)( 53,206)( 54,203)( 55,201)( 56,202)( 57,145)
( 58,146)( 59,144)( 60,141)( 61,142)( 62,143)( 63,140)( 64,138)( 65,139)
( 66,163)( 67,164)( 68,162)( 69,159)( 70,160)( 71,161)( 72,158)( 73,156)
( 74,157)( 75,154)( 76,155)( 77,153)( 78,150)( 79,151)( 80,152)( 81,149)
( 82,147)( 83,148)( 84,118)( 85,119)( 86,117)( 87,114)( 88,115)( 89,116)
( 90,113)( 91,111)( 92,112)( 93,136)( 94,137)( 95,135)( 96,132)( 97,133)
( 98,134)( 99,131)(100,129)(101,130)(102,127)(103,128)(104,126)(105,123)
(106,124)(107,125)(108,122)(109,120)(110,121)(219,388)(220,389)(221,387)
(222,384)(223,385)(224,386)(225,383)(226,381)(227,382)(228,406)(229,407)
(230,405)(231,402)(232,403)(233,404)(234,401)(235,399)(236,400)(237,397)
(238,398)(239,396)(240,393)(241,394)(242,395)(243,392)(244,390)(245,391)
(246,415)(247,416)(248,414)(249,411)(250,412)(251,413)(252,410)(253,408)
(254,409)(255,433)(256,434)(257,432)(258,429)(259,430)(260,431)(261,428)
(262,426)(263,427)(264,424)(265,425)(266,423)(267,420)(268,421)(269,422)
(270,419)(271,417)(272,418)(273,361)(274,362)(275,360)(276,357)(277,358)
(278,359)(279,356)(280,354)(281,355)(282,379)(283,380)(284,378)(285,375)
(286,376)(287,377)(288,374)(289,372)(290,373)(291,370)(292,371)(293,369)
(294,366)(295,367)(296,368)(297,365)(298,363)(299,364)(300,334)(301,335)
(302,333)(303,330)(304,331)(305,332)(306,329)(307,327)(308,328)(309,352)
(310,353)(311,351)(312,348)(313,349)(314,350)(315,347)(316,345)(317,346)
(318,343)(319,344)(320,342)(321,339)(322,340)(323,341)(324,338)(325,336)
(326,337);
poly := sub<Sym(434)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3 >;
to this polytope