Polytope of Type {78,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78,12}*1872b
Also Known As : {78,12|2}. if this polytope has another name.
Group : SmallGroup(1872,907)
Rank : 3
Schlafli Type : {78,12}
Number of vertices, edges, etc : 78, 468, 12
Order of s0s1s2 : 156
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {78,6}*936b
3-fold quotients : {26,12}*624, {78,4}*624a
6-fold quotients : {26,6}*312, {78,2}*312
9-fold quotients : {26,4}*208
12-fold quotients : {39,2}*156
13-fold quotients : {6,12}*144a
18-fold quotients : {26,2}*104
26-fold quotients : {6,6}*72a
36-fold quotients : {13,2}*52
39-fold quotients : {2,12}*48, {6,4}*48a
78-fold quotients : {2,6}*24, {6,2}*24
117-fold quotients : {2,4}*16
156-fold quotients : {2,3}*12, {3,2}*12
234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 13)( 3, 12)( 4, 11)( 5, 10)( 6, 9)( 7, 8)( 14, 27)( 15, 39)
( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)
( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 80, 91)( 81, 90)
( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)( 95,115)
( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)(103,107)
(104,106)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(131,144)
(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)
(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)(161,166)
(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)(175,191)
(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(197,208)
(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)(210,234)(211,233)
(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)(218,226)(219,225)
(220,224)(221,223)(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)
(248,261)(249,273)(250,272)(251,271)(252,270)(253,269)(254,268)(255,267)
(256,266)(257,265)(258,264)(259,263)(260,262)(275,286)(276,285)(277,284)
(278,283)(279,282)(280,281)(287,300)(288,312)(289,311)(290,310)(291,309)
(292,308)(293,307)(294,306)(295,305)(296,304)(297,303)(298,302)(299,301)
(314,325)(315,324)(316,323)(317,322)(318,321)(319,320)(326,339)(327,351)
(328,350)(329,349)(330,348)(331,347)(332,346)(333,345)(334,344)(335,343)
(336,342)(337,341)(338,340)(353,364)(354,363)(355,362)(356,361)(357,360)
(358,359)(365,378)(366,390)(367,389)(368,388)(369,387)(370,386)(371,385)
(372,384)(373,383)(374,382)(375,381)(376,380)(377,379)(392,403)(393,402)
(394,401)(395,400)(396,399)(397,398)(404,417)(405,429)(406,428)(407,427)
(408,426)(409,425)(410,424)(411,423)(412,422)(413,421)(414,420)(415,419)
(416,418)(431,442)(432,441)(433,440)(434,439)(435,438)(436,437)(443,456)
(444,468)(445,467)(446,466)(447,465)(448,464)(449,463)(450,462)(451,461)
(452,460)(453,459)(454,458)(455,457);;
s1 := ( 1, 15)( 2, 14)( 3, 26)( 4, 25)( 5, 24)( 6, 23)( 7, 22)( 8, 21)
( 9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)( 44,102)
( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)( 52, 94)
( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)( 60, 86)
( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)( 68,117)
( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,108)( 78,107)(118,132)(119,131)(120,143)(121,142)(122,141)(123,140)
(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(144,145)
(146,156)(147,155)(148,154)(149,153)(150,152)(157,210)(158,209)(159,221)
(160,220)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)(167,213)
(168,212)(169,211)(170,197)(171,196)(172,208)(173,207)(174,206)(175,205)
(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)(183,223)
(184,222)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)(191,228)
(192,227)(193,226)(194,225)(195,224)(235,366)(236,365)(237,377)(238,376)
(239,375)(240,374)(241,373)(242,372)(243,371)(244,370)(245,369)(246,368)
(247,367)(248,353)(249,352)(250,364)(251,363)(252,362)(253,361)(254,360)
(255,359)(256,358)(257,357)(258,356)(259,355)(260,354)(261,379)(262,378)
(263,390)(264,389)(265,388)(266,387)(267,386)(268,385)(269,384)(270,383)
(271,382)(272,381)(273,380)(274,444)(275,443)(276,455)(277,454)(278,453)
(279,452)(280,451)(281,450)(282,449)(283,448)(284,447)(285,446)(286,445)
(287,431)(288,430)(289,442)(290,441)(291,440)(292,439)(293,438)(294,437)
(295,436)(296,435)(297,434)(298,433)(299,432)(300,457)(301,456)(302,468)
(303,467)(304,466)(305,465)(306,464)(307,463)(308,462)(309,461)(310,460)
(311,459)(312,458)(313,405)(314,404)(315,416)(316,415)(317,414)(318,413)
(319,412)(320,411)(321,410)(322,409)(323,408)(324,407)(325,406)(326,392)
(327,391)(328,403)(329,402)(330,401)(331,400)(332,399)(333,398)(334,397)
(335,396)(336,395)(337,394)(338,393)(339,418)(340,417)(341,429)(342,428)
(343,427)(344,426)(345,425)(346,424)(347,423)(348,422)(349,421)(350,420)
(351,419);;
s2 := ( 1,274)( 2,275)( 3,276)( 4,277)( 5,278)( 6,279)( 7,280)( 8,281)
( 9,282)( 10,283)( 11,284)( 12,285)( 13,286)( 14,287)( 15,288)( 16,289)
( 17,290)( 18,291)( 19,292)( 20,293)( 21,294)( 22,295)( 23,296)( 24,297)
( 25,298)( 26,299)( 27,300)( 28,301)( 29,302)( 30,303)( 31,304)( 32,305)
( 33,306)( 34,307)( 35,308)( 36,309)( 37,310)( 38,311)( 39,312)( 40,235)
( 41,236)( 42,237)( 43,238)( 44,239)( 45,240)( 46,241)( 47,242)( 48,243)
( 49,244)( 50,245)( 51,246)( 52,247)( 53,248)( 54,249)( 55,250)( 56,251)
( 57,252)( 58,253)( 59,254)( 60,255)( 61,256)( 62,257)( 63,258)( 64,259)
( 65,260)( 66,261)( 67,262)( 68,263)( 69,264)( 70,265)( 71,266)( 72,267)
( 73,268)( 74,269)( 75,270)( 76,271)( 77,272)( 78,273)( 79,313)( 80,314)
( 81,315)( 82,316)( 83,317)( 84,318)( 85,319)( 86,320)( 87,321)( 88,322)
( 89,323)( 90,324)( 91,325)( 92,326)( 93,327)( 94,328)( 95,329)( 96,330)
( 97,331)( 98,332)( 99,333)(100,334)(101,335)(102,336)(103,337)(104,338)
(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)(112,346)
(113,347)(114,348)(115,349)(116,350)(117,351)(118,391)(119,392)(120,393)
(121,394)(122,395)(123,396)(124,397)(125,398)(126,399)(127,400)(128,401)
(129,402)(130,403)(131,404)(132,405)(133,406)(134,407)(135,408)(136,409)
(137,410)(138,411)(139,412)(140,413)(141,414)(142,415)(143,416)(144,417)
(145,418)(146,419)(147,420)(148,421)(149,422)(150,423)(151,424)(152,425)
(153,426)(154,427)(155,428)(156,429)(157,352)(158,353)(159,354)(160,355)
(161,356)(162,357)(163,358)(164,359)(165,360)(166,361)(167,362)(168,363)
(169,364)(170,365)(171,366)(172,367)(173,368)(174,369)(175,370)(176,371)
(177,372)(178,373)(179,374)(180,375)(181,376)(182,377)(183,378)(184,379)
(185,380)(186,381)(187,382)(188,383)(189,384)(190,385)(191,386)(192,387)
(193,388)(194,389)(195,390)(196,430)(197,431)(198,432)(199,433)(200,434)
(201,435)(202,436)(203,437)(204,438)(205,439)(206,440)(207,441)(208,442)
(209,443)(210,444)(211,445)(212,446)(213,447)(214,448)(215,449)(216,450)
(217,451)(218,452)(219,453)(220,454)(221,455)(222,456)(223,457)(224,458)
(225,459)(226,460)(227,461)(228,462)(229,463)(230,464)(231,465)(232,466)
(233,467)(234,468);;
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*s2*s1*s0*s1*s2*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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(468)!( 2, 13)( 3, 12)( 4, 11)( 5, 10)( 6, 9)( 7, 8)( 14, 27)
( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)
( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)
( 45, 48)( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 80, 91)
( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)
( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)
(103,107)(104,106)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)
(131,144)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)
(139,149)(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)
(161,166)(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)
(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)
(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)(210,234)
(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)(218,226)
(219,225)(220,224)(221,223)(236,247)(237,246)(238,245)(239,244)(240,243)
(241,242)(248,261)(249,273)(250,272)(251,271)(252,270)(253,269)(254,268)
(255,267)(256,266)(257,265)(258,264)(259,263)(260,262)(275,286)(276,285)
(277,284)(278,283)(279,282)(280,281)(287,300)(288,312)(289,311)(290,310)
(291,309)(292,308)(293,307)(294,306)(295,305)(296,304)(297,303)(298,302)
(299,301)(314,325)(315,324)(316,323)(317,322)(318,321)(319,320)(326,339)
(327,351)(328,350)(329,349)(330,348)(331,347)(332,346)(333,345)(334,344)
(335,343)(336,342)(337,341)(338,340)(353,364)(354,363)(355,362)(356,361)
(357,360)(358,359)(365,378)(366,390)(367,389)(368,388)(369,387)(370,386)
(371,385)(372,384)(373,383)(374,382)(375,381)(376,380)(377,379)(392,403)
(393,402)(394,401)(395,400)(396,399)(397,398)(404,417)(405,429)(406,428)
(407,427)(408,426)(409,425)(410,424)(411,423)(412,422)(413,421)(414,420)
(415,419)(416,418)(431,442)(432,441)(433,440)(434,439)(435,438)(436,437)
(443,456)(444,468)(445,467)(446,466)(447,465)(448,464)(449,463)(450,462)
(451,461)(452,460)(453,459)(454,458)(455,457);
s1 := Sym(468)!( 1, 15)( 2, 14)( 3, 26)( 4, 25)( 5, 24)( 6, 23)( 7, 22)
( 8, 21)( 9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)
( 30, 38)( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)
( 44,102)( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)
( 52, 94)( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)
( 60, 86)( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)
( 68,117)( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)
( 76,109)( 77,108)( 78,107)(118,132)(119,131)(120,143)(121,142)(122,141)
(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)
(144,145)(146,156)(147,155)(148,154)(149,153)(150,152)(157,210)(158,209)
(159,221)(160,220)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)
(167,213)(168,212)(169,211)(170,197)(171,196)(172,208)(173,207)(174,206)
(175,205)(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)
(183,223)(184,222)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)
(191,228)(192,227)(193,226)(194,225)(195,224)(235,366)(236,365)(237,377)
(238,376)(239,375)(240,374)(241,373)(242,372)(243,371)(244,370)(245,369)
(246,368)(247,367)(248,353)(249,352)(250,364)(251,363)(252,362)(253,361)
(254,360)(255,359)(256,358)(257,357)(258,356)(259,355)(260,354)(261,379)
(262,378)(263,390)(264,389)(265,388)(266,387)(267,386)(268,385)(269,384)
(270,383)(271,382)(272,381)(273,380)(274,444)(275,443)(276,455)(277,454)
(278,453)(279,452)(280,451)(281,450)(282,449)(283,448)(284,447)(285,446)
(286,445)(287,431)(288,430)(289,442)(290,441)(291,440)(292,439)(293,438)
(294,437)(295,436)(296,435)(297,434)(298,433)(299,432)(300,457)(301,456)
(302,468)(303,467)(304,466)(305,465)(306,464)(307,463)(308,462)(309,461)
(310,460)(311,459)(312,458)(313,405)(314,404)(315,416)(316,415)(317,414)
(318,413)(319,412)(320,411)(321,410)(322,409)(323,408)(324,407)(325,406)
(326,392)(327,391)(328,403)(329,402)(330,401)(331,400)(332,399)(333,398)
(334,397)(335,396)(336,395)(337,394)(338,393)(339,418)(340,417)(341,429)
(342,428)(343,427)(344,426)(345,425)(346,424)(347,423)(348,422)(349,421)
(350,420)(351,419);
s2 := Sym(468)!( 1,274)( 2,275)( 3,276)( 4,277)( 5,278)( 6,279)( 7,280)
( 8,281)( 9,282)( 10,283)( 11,284)( 12,285)( 13,286)( 14,287)( 15,288)
( 16,289)( 17,290)( 18,291)( 19,292)( 20,293)( 21,294)( 22,295)( 23,296)
( 24,297)( 25,298)( 26,299)( 27,300)( 28,301)( 29,302)( 30,303)( 31,304)
( 32,305)( 33,306)( 34,307)( 35,308)( 36,309)( 37,310)( 38,311)( 39,312)
( 40,235)( 41,236)( 42,237)( 43,238)( 44,239)( 45,240)( 46,241)( 47,242)
( 48,243)( 49,244)( 50,245)( 51,246)( 52,247)( 53,248)( 54,249)( 55,250)
( 56,251)( 57,252)( 58,253)( 59,254)( 60,255)( 61,256)( 62,257)( 63,258)
( 64,259)( 65,260)( 66,261)( 67,262)( 68,263)( 69,264)( 70,265)( 71,266)
( 72,267)( 73,268)( 74,269)( 75,270)( 76,271)( 77,272)( 78,273)( 79,313)
( 80,314)( 81,315)( 82,316)( 83,317)( 84,318)( 85,319)( 86,320)( 87,321)
( 88,322)( 89,323)( 90,324)( 91,325)( 92,326)( 93,327)( 94,328)( 95,329)
( 96,330)( 97,331)( 98,332)( 99,333)(100,334)(101,335)(102,336)(103,337)
(104,338)(105,339)(106,340)(107,341)(108,342)(109,343)(110,344)(111,345)
(112,346)(113,347)(114,348)(115,349)(116,350)(117,351)(118,391)(119,392)
(120,393)(121,394)(122,395)(123,396)(124,397)(125,398)(126,399)(127,400)
(128,401)(129,402)(130,403)(131,404)(132,405)(133,406)(134,407)(135,408)
(136,409)(137,410)(138,411)(139,412)(140,413)(141,414)(142,415)(143,416)
(144,417)(145,418)(146,419)(147,420)(148,421)(149,422)(150,423)(151,424)
(152,425)(153,426)(154,427)(155,428)(156,429)(157,352)(158,353)(159,354)
(160,355)(161,356)(162,357)(163,358)(164,359)(165,360)(166,361)(167,362)
(168,363)(169,364)(170,365)(171,366)(172,367)(173,368)(174,369)(175,370)
(176,371)(177,372)(178,373)(179,374)(180,375)(181,376)(182,377)(183,378)
(184,379)(185,380)(186,381)(187,382)(188,383)(189,384)(190,385)(191,386)
(192,387)(193,388)(194,389)(195,390)(196,430)(197,431)(198,432)(199,433)
(200,434)(201,435)(202,436)(203,437)(204,438)(205,439)(206,440)(207,441)
(208,442)(209,443)(210,444)(211,445)(212,446)(213,447)(214,448)(215,449)
(216,450)(217,451)(218,452)(219,453)(220,454)(221,455)(222,456)(223,457)
(224,458)(225,459)(226,460)(227,461)(228,462)(229,463)(230,464)(231,465)
(232,466)(233,467)(234,468);
poly := sub<Sym(468)|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*s2*s1*s0*s1*s2*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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope