Polytope of Type {78,10}

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