Polytope of Type {2,8,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30374)
Rank : 4
Schlafli Type : {2,8,6}
Number of vertices, edges, etc : 2, 72, 216, 54
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 : {2,4,6}*864a
3-fold quotients : {2,8,6}*576
4-fold quotients : {2,4,6}*432
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 := ( 3,219)( 4,221)( 5,220)( 6,230)( 7,229)( 8,228)( 9,238)( 10,237)
( 11,239)( 12,224)( 13,223)( 14,222)( 15,233)( 16,232)( 17,231)( 18,242)
( 19,241)( 20,240)( 21,226)( 22,225)( 23,227)( 24,236)( 25,235)( 26,234)
( 27,243)( 28,245)( 29,244)( 30,246)( 31,248)( 32,247)( 33,257)( 34,256)
( 35,255)( 36,265)( 37,264)( 38,266)( 39,251)( 40,250)( 41,249)( 42,260)
( 43,259)( 44,258)( 45,269)( 46,268)( 47,267)( 48,253)( 49,252)( 50,254)
( 51,263)( 52,262)( 53,261)( 54,270)( 55,272)( 56,271)( 57,300)( 58,302)
( 59,301)( 60,311)( 61,310)( 62,309)( 63,319)( 64,318)( 65,320)( 66,305)
( 67,304)( 68,303)( 69,314)( 70,313)( 71,312)( 72,323)( 73,322)( 74,321)
( 75,307)( 76,306)( 77,308)( 78,317)( 79,316)( 80,315)( 81,324)( 82,326)
( 83,325)( 84,273)( 85,275)( 86,274)( 87,284)( 88,283)( 89,282)( 90,292)
( 91,291)( 92,293)( 93,278)( 94,277)( 95,276)( 96,287)( 97,286)( 98,285)
( 99,296)(100,295)(101,294)(102,280)(103,279)(104,281)(105,290)(106,289)
(107,288)(108,297)(109,299)(110,298)(111,354)(112,356)(113,355)(114,365)
(115,364)(116,363)(117,373)(118,372)(119,374)(120,359)(121,358)(122,357)
(123,368)(124,367)(125,366)(126,377)(127,376)(128,375)(129,361)(130,360)
(131,362)(132,371)(133,370)(134,369)(135,378)(136,380)(137,379)(138,327)
(139,329)(140,328)(141,338)(142,337)(143,336)(144,346)(145,345)(146,347)
(147,332)(148,331)(149,330)(150,341)(151,340)(152,339)(153,350)(154,349)
(155,348)(156,334)(157,333)(158,335)(159,344)(160,343)(161,342)(162,351)
(163,353)(164,352)(165,381)(166,383)(167,382)(168,392)(169,391)(170,390)
(171,400)(172,399)(173,401)(174,386)(175,385)(176,384)(177,395)(178,394)
(179,393)(180,404)(181,403)(182,402)(183,388)(184,387)(185,389)(186,398)
(187,397)(188,396)(189,405)(190,407)(191,406)(192,408)(193,410)(194,409)
(195,419)(196,418)(197,417)(198,427)(199,426)(200,428)(201,413)(202,412)
(203,411)(204,422)(205,421)(206,420)(207,431)(208,430)(209,429)(210,415)
(211,414)(212,416)(213,425)(214,424)(215,423)(216,432)(217,434)(218,433);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2 ];;
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)!( 3,219)( 4,221)( 5,220)( 6,230)( 7,229)( 8,228)( 9,238)
( 10,237)( 11,239)( 12,224)( 13,223)( 14,222)( 15,233)( 16,232)( 17,231)
( 18,242)( 19,241)( 20,240)( 21,226)( 22,225)( 23,227)( 24,236)( 25,235)
( 26,234)( 27,243)( 28,245)( 29,244)( 30,246)( 31,248)( 32,247)( 33,257)
( 34,256)( 35,255)( 36,265)( 37,264)( 38,266)( 39,251)( 40,250)( 41,249)
( 42,260)( 43,259)( 44,258)( 45,269)( 46,268)( 47,267)( 48,253)( 49,252)
( 50,254)( 51,263)( 52,262)( 53,261)( 54,270)( 55,272)( 56,271)( 57,300)
( 58,302)( 59,301)( 60,311)( 61,310)( 62,309)( 63,319)( 64,318)( 65,320)
( 66,305)( 67,304)( 68,303)( 69,314)( 70,313)( 71,312)( 72,323)( 73,322)
( 74,321)( 75,307)( 76,306)( 77,308)( 78,317)( 79,316)( 80,315)( 81,324)
( 82,326)( 83,325)( 84,273)( 85,275)( 86,274)( 87,284)( 88,283)( 89,282)
( 90,292)( 91,291)( 92,293)( 93,278)( 94,277)( 95,276)( 96,287)( 97,286)
( 98,285)( 99,296)(100,295)(101,294)(102,280)(103,279)(104,281)(105,290)
(106,289)(107,288)(108,297)(109,299)(110,298)(111,354)(112,356)(113,355)
(114,365)(115,364)(116,363)(117,373)(118,372)(119,374)(120,359)(121,358)
(122,357)(123,368)(124,367)(125,366)(126,377)(127,376)(128,375)(129,361)
(130,360)(131,362)(132,371)(133,370)(134,369)(135,378)(136,380)(137,379)
(138,327)(139,329)(140,328)(141,338)(142,337)(143,336)(144,346)(145,345)
(146,347)(147,332)(148,331)(149,330)(150,341)(151,340)(152,339)(153,350)
(154,349)(155,348)(156,334)(157,333)(158,335)(159,344)(160,343)(161,342)
(162,351)(163,353)(164,352)(165,381)(166,383)(167,382)(168,392)(169,391)
(170,390)(171,400)(172,399)(173,401)(174,386)(175,385)(176,384)(177,395)
(178,394)(179,393)(180,404)(181,403)(182,402)(183,388)(184,387)(185,389)
(186,398)(187,397)(188,396)(189,405)(190,407)(191,406)(192,408)(193,410)
(194,409)(195,419)(196,418)(197,417)(198,427)(199,426)(200,428)(201,413)
(202,412)(203,411)(204,422)(205,421)(206,420)(207,431)(208,430)(209,429)
(210,415)(211,414)(212,416)(213,425)(214,424)(215,423)(216,432)(217,434)
(218,433);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2 >;
to this polytope