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Polytope of Type {10,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,78}*1560
Also Known As : {10,78|2}. if this polytope has another name.
Group : SmallGroup(1560,208)
Rank : 3
Schlafli Type : {10,78}
Number of vertices, edges, etc : 10, 390, 78
Order of s0s1s2 : 390
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) :
   3-fold quotients : {10,26}*520
   5-fold quotients : {2,78}*312
   10-fold quotients : {2,39}*156
   13-fold quotients : {10,6}*120
   15-fold quotients : {2,26}*104
   30-fold quotients : {2,13}*52
   39-fold quotients : {10,2}*40
   65-fold quotients : {2,6}*24
   78-fold quotients : {5,2}*20
   130-fold quotients : {2,3}*12
   195-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 14, 53)( 15, 54)( 16, 55)( 17, 56)( 18, 57)( 19, 58)( 20, 59)( 21, 60)
( 22, 61)( 23, 62)( 24, 63)( 25, 64)( 26, 65)( 27, 40)( 28, 41)( 29, 42)
( 30, 43)( 31, 44)( 32, 45)( 33, 46)( 34, 47)( 35, 48)( 36, 49)( 37, 50)
( 38, 51)( 39, 52)( 79,118)( 80,119)( 81,120)( 82,121)( 83,122)( 84,123)
( 85,124)( 86,125)( 87,126)( 88,127)( 89,128)( 90,129)( 91,130)( 92,105)
( 93,106)( 94,107)( 95,108)( 96,109)( 97,110)( 98,111)( 99,112)(100,113)
(101,114)(102,115)(103,116)(104,117)(144,183)(145,184)(146,185)(147,186)
(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)
(156,195)(157,170)(158,171)(159,172)(160,173)(161,174)(162,175)(163,176)
(164,177)(165,178)(166,179)(167,180)(168,181)(169,182)(209,248)(210,249)
(211,250)(212,251)(213,252)(214,253)(215,254)(216,255)(217,256)(218,257)
(219,258)(220,259)(221,260)(222,235)(223,236)(224,237)(225,238)(226,239)
(227,240)(228,241)(229,242)(230,243)(231,244)(232,245)(233,246)(234,247)
(274,313)(275,314)(276,315)(277,316)(278,317)(279,318)(280,319)(281,320)
(282,321)(283,322)(284,323)(285,324)(286,325)(287,300)(288,301)(289,302)
(290,303)(291,304)(292,305)(293,306)(294,307)(295,308)(296,309)(297,310)
(298,311)(299,312)(339,378)(340,379)(341,380)(342,381)(343,382)(344,383)
(345,384)(346,385)(347,386)(348,387)(349,388)(350,389)(351,390)(352,365)
(353,366)(354,367)(355,368)(356,369)(357,370)(358,371)(359,372)(360,373)
(361,374)(362,375)(363,376)(364,377);;
s1 := (  1, 14)(  2, 26)(  3, 25)(  4, 24)(  5, 23)(  6, 22)(  7, 21)(  8, 20)
(  9, 19)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 27, 53)( 28, 65)( 29, 64)
( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)
( 38, 55)( 39, 54)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)( 46, 47)
( 66,144)( 67,156)( 68,155)( 69,154)( 70,153)( 71,152)( 72,151)( 73,150)
( 74,149)( 75,148)( 76,147)( 77,146)( 78,145)( 79,131)( 80,143)( 81,142)
( 82,141)( 83,140)( 84,139)( 85,138)( 86,137)( 87,136)( 88,135)( 89,134)
( 90,133)( 91,132)( 92,183)( 93,195)( 94,194)( 95,193)( 96,192)( 97,191)
( 98,190)( 99,189)(100,188)(101,187)(102,186)(103,185)(104,184)(105,170)
(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176)(113,175)
(114,174)(115,173)(116,172)(117,171)(118,157)(119,169)(120,168)(121,167)
(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)
(130,158)(196,209)(197,221)(198,220)(199,219)(200,218)(201,217)(202,216)
(203,215)(204,214)(205,213)(206,212)(207,211)(208,210)(222,248)(223,260)
(224,259)(225,258)(226,257)(227,256)(228,255)(229,254)(230,253)(231,252)
(232,251)(233,250)(234,249)(236,247)(237,246)(238,245)(239,244)(240,243)
(241,242)(261,339)(262,351)(263,350)(264,349)(265,348)(266,347)(267,346)
(268,345)(269,344)(270,343)(271,342)(272,341)(273,340)(274,326)(275,338)
(276,337)(277,336)(278,335)(279,334)(280,333)(281,332)(282,331)(283,330)
(284,329)(285,328)(286,327)(287,378)(288,390)(289,389)(290,388)(291,387)
(292,386)(293,385)(294,384)(295,383)(296,382)(297,381)(298,380)(299,379)
(300,365)(301,377)(302,376)(303,375)(304,374)(305,373)(306,372)(307,371)
(308,370)(309,369)(310,368)(311,367)(312,366)(313,352)(314,364)(315,363)
(316,362)(317,361)(318,360)(319,359)(320,358)(321,357)(322,356)(323,355)
(324,354)(325,353);;
s2 := (  1,262)(  2,261)(  3,273)(  4,272)(  5,271)(  6,270)(  7,269)(  8,268)
(  9,267)( 10,266)( 11,265)( 12,264)( 13,263)( 14,275)( 15,274)( 16,286)
( 17,285)( 18,284)( 19,283)( 20,282)( 21,281)( 22,280)( 23,279)( 24,278)
( 25,277)( 26,276)( 27,288)( 28,287)( 29,299)( 30,298)( 31,297)( 32,296)
( 33,295)( 34,294)( 35,293)( 36,292)( 37,291)( 38,290)( 39,289)( 40,301)
( 41,300)( 42,312)( 43,311)( 44,310)( 45,309)( 46,308)( 47,307)( 48,306)
( 49,305)( 50,304)( 51,303)( 52,302)( 53,314)( 54,313)( 55,325)( 56,324)
( 57,323)( 58,322)( 59,321)( 60,320)( 61,319)( 62,318)( 63,317)( 64,316)
( 65,315)( 66,197)( 67,196)( 68,208)( 69,207)( 70,206)( 71,205)( 72,204)
( 73,203)( 74,202)( 75,201)( 76,200)( 77,199)( 78,198)( 79,210)( 80,209)
( 81,221)( 82,220)( 83,219)( 84,218)( 85,217)( 86,216)( 87,215)( 88,214)
( 89,213)( 90,212)( 91,211)( 92,223)( 93,222)( 94,234)( 95,233)( 96,232)
( 97,231)( 98,230)( 99,229)(100,228)(101,227)(102,226)(103,225)(104,224)
(105,236)(106,235)(107,247)(108,246)(109,245)(110,244)(111,243)(112,242)
(113,241)(114,240)(115,239)(116,238)(117,237)(118,249)(119,248)(120,260)
(121,259)(122,258)(123,257)(124,256)(125,255)(126,254)(127,253)(128,252)
(129,251)(130,250)(131,327)(132,326)(133,338)(134,337)(135,336)(136,335)
(137,334)(138,333)(139,332)(140,331)(141,330)(142,329)(143,328)(144,340)
(145,339)(146,351)(147,350)(148,349)(149,348)(150,347)(151,346)(152,345)
(153,344)(154,343)(155,342)(156,341)(157,353)(158,352)(159,364)(160,363)
(161,362)(162,361)(163,360)(164,359)(165,358)(166,357)(167,356)(168,355)
(169,354)(170,366)(171,365)(172,377)(173,376)(174,375)(175,374)(176,373)
(177,372)(178,371)(179,370)(180,369)(181,368)(182,367)(183,379)(184,378)
(185,390)(186,389)(187,388)(188,387)(189,386)(190,385)(191,384)(192,383)
(193,382)(194,381)(195,380);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*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(390)!( 14, 53)( 15, 54)( 16, 55)( 17, 56)( 18, 57)( 19, 58)( 20, 59)
( 21, 60)( 22, 61)( 23, 62)( 24, 63)( 25, 64)( 26, 65)( 27, 40)( 28, 41)
( 29, 42)( 30, 43)( 31, 44)( 32, 45)( 33, 46)( 34, 47)( 35, 48)( 36, 49)
( 37, 50)( 38, 51)( 39, 52)( 79,118)( 80,119)( 81,120)( 82,121)( 83,122)
( 84,123)( 85,124)( 86,125)( 87,126)( 88,127)( 89,128)( 90,129)( 91,130)
( 92,105)( 93,106)( 94,107)( 95,108)( 96,109)( 97,110)( 98,111)( 99,112)
(100,113)(101,114)(102,115)(103,116)(104,117)(144,183)(145,184)(146,185)
(147,186)(148,187)(149,188)(150,189)(151,190)(152,191)(153,192)(154,193)
(155,194)(156,195)(157,170)(158,171)(159,172)(160,173)(161,174)(162,175)
(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,182)(209,248)
(210,249)(211,250)(212,251)(213,252)(214,253)(215,254)(216,255)(217,256)
(218,257)(219,258)(220,259)(221,260)(222,235)(223,236)(224,237)(225,238)
(226,239)(227,240)(228,241)(229,242)(230,243)(231,244)(232,245)(233,246)
(234,247)(274,313)(275,314)(276,315)(277,316)(278,317)(279,318)(280,319)
(281,320)(282,321)(283,322)(284,323)(285,324)(286,325)(287,300)(288,301)
(289,302)(290,303)(291,304)(292,305)(293,306)(294,307)(295,308)(296,309)
(297,310)(298,311)(299,312)(339,378)(340,379)(341,380)(342,381)(343,382)
(344,383)(345,384)(346,385)(347,386)(348,387)(349,388)(350,389)(351,390)
(352,365)(353,366)(354,367)(355,368)(356,369)(357,370)(358,371)(359,372)
(360,373)(361,374)(362,375)(363,376)(364,377);
s1 := Sym(390)!(  1, 14)(  2, 26)(  3, 25)(  4, 24)(  5, 23)(  6, 22)(  7, 21)
(  8, 20)(  9, 19)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 27, 53)( 28, 65)
( 29, 64)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 57)
( 37, 56)( 38, 55)( 39, 54)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 66,144)( 67,156)( 68,155)( 69,154)( 70,153)( 71,152)( 72,151)
( 73,150)( 74,149)( 75,148)( 76,147)( 77,146)( 78,145)( 79,131)( 80,143)
( 81,142)( 82,141)( 83,140)( 84,139)( 85,138)( 86,137)( 87,136)( 88,135)
( 89,134)( 90,133)( 91,132)( 92,183)( 93,195)( 94,194)( 95,193)( 96,192)
( 97,191)( 98,190)( 99,189)(100,188)(101,187)(102,186)(103,185)(104,184)
(105,170)(106,182)(107,181)(108,180)(109,179)(110,178)(111,177)(112,176)
(113,175)(114,174)(115,173)(116,172)(117,171)(118,157)(119,169)(120,168)
(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)
(129,159)(130,158)(196,209)(197,221)(198,220)(199,219)(200,218)(201,217)
(202,216)(203,215)(204,214)(205,213)(206,212)(207,211)(208,210)(222,248)
(223,260)(224,259)(225,258)(226,257)(227,256)(228,255)(229,254)(230,253)
(231,252)(232,251)(233,250)(234,249)(236,247)(237,246)(238,245)(239,244)
(240,243)(241,242)(261,339)(262,351)(263,350)(264,349)(265,348)(266,347)
(267,346)(268,345)(269,344)(270,343)(271,342)(272,341)(273,340)(274,326)
(275,338)(276,337)(277,336)(278,335)(279,334)(280,333)(281,332)(282,331)
(283,330)(284,329)(285,328)(286,327)(287,378)(288,390)(289,389)(290,388)
(291,387)(292,386)(293,385)(294,384)(295,383)(296,382)(297,381)(298,380)
(299,379)(300,365)(301,377)(302,376)(303,375)(304,374)(305,373)(306,372)
(307,371)(308,370)(309,369)(310,368)(311,367)(312,366)(313,352)(314,364)
(315,363)(316,362)(317,361)(318,360)(319,359)(320,358)(321,357)(322,356)
(323,355)(324,354)(325,353);
s2 := Sym(390)!(  1,262)(  2,261)(  3,273)(  4,272)(  5,271)(  6,270)(  7,269)
(  8,268)(  9,267)( 10,266)( 11,265)( 12,264)( 13,263)( 14,275)( 15,274)
( 16,286)( 17,285)( 18,284)( 19,283)( 20,282)( 21,281)( 22,280)( 23,279)
( 24,278)( 25,277)( 26,276)( 27,288)( 28,287)( 29,299)( 30,298)( 31,297)
( 32,296)( 33,295)( 34,294)( 35,293)( 36,292)( 37,291)( 38,290)( 39,289)
( 40,301)( 41,300)( 42,312)( 43,311)( 44,310)( 45,309)( 46,308)( 47,307)
( 48,306)( 49,305)( 50,304)( 51,303)( 52,302)( 53,314)( 54,313)( 55,325)
( 56,324)( 57,323)( 58,322)( 59,321)( 60,320)( 61,319)( 62,318)( 63,317)
( 64,316)( 65,315)( 66,197)( 67,196)( 68,208)( 69,207)( 70,206)( 71,205)
( 72,204)( 73,203)( 74,202)( 75,201)( 76,200)( 77,199)( 78,198)( 79,210)
( 80,209)( 81,221)( 82,220)( 83,219)( 84,218)( 85,217)( 86,216)( 87,215)
( 88,214)( 89,213)( 90,212)( 91,211)( 92,223)( 93,222)( 94,234)( 95,233)
( 96,232)( 97,231)( 98,230)( 99,229)(100,228)(101,227)(102,226)(103,225)
(104,224)(105,236)(106,235)(107,247)(108,246)(109,245)(110,244)(111,243)
(112,242)(113,241)(114,240)(115,239)(116,238)(117,237)(118,249)(119,248)
(120,260)(121,259)(122,258)(123,257)(124,256)(125,255)(126,254)(127,253)
(128,252)(129,251)(130,250)(131,327)(132,326)(133,338)(134,337)(135,336)
(136,335)(137,334)(138,333)(139,332)(140,331)(141,330)(142,329)(143,328)
(144,340)(145,339)(146,351)(147,350)(148,349)(149,348)(150,347)(151,346)
(152,345)(153,344)(154,343)(155,342)(156,341)(157,353)(158,352)(159,364)
(160,363)(161,362)(162,361)(163,360)(164,359)(165,358)(166,357)(167,356)
(168,355)(169,354)(170,366)(171,365)(172,377)(173,376)(174,375)(175,374)
(176,373)(177,372)(178,371)(179,370)(180,369)(181,368)(182,367)(183,379)
(184,378)(185,390)(186,389)(187,388)(188,387)(189,386)(190,385)(191,384)
(192,383)(193,382)(194,381)(195,380);
poly := sub<Sym(390)|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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope