Polytope of Type {54,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,14}*1512
Also Known As : {54,14|2}. if this polytope has another name.
Group : SmallGroup(1512,89)
Rank : 3
Schlafli Type : {54,14}
Number of vertices, edges, etc : 54, 378, 14
Order of s₀s₁s₂ : 378
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 : {18,14}*504
   7-fold quotients : {54,2}*216
   9-fold quotients : {6,14}*168
   14-fold quotients : {27,2}*108
   21-fold quotients : {18,2}*72
   27-fold quotients : {2,14}*56
   42-fold quotients : {9,2}*36
   54-fold quotients : {2,7}*28
   63-fold quotients : {6,2}*24
   126-fold quotients : {3,2}*12
   189-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)( 15, 18)
( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)( 33, 36)
( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)( 51, 54)
( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64,131)( 65,130)( 66,132)( 67,128)
( 68,127)( 69,129)( 70,135)( 71,134)( 72,133)( 73,140)( 74,139)( 75,141)
( 76,137)( 77,136)( 78,138)( 79,144)( 80,143)( 81,142)( 82,149)( 83,148)
( 84,150)( 85,146)( 86,145)( 87,147)( 88,153)( 89,152)( 90,151)( 91,158)
( 92,157)( 93,159)( 94,155)( 95,154)( 96,156)( 97,162)( 98,161)( 99,160)
(100,167)(101,166)(102,168)(103,164)(104,163)(105,165)(106,171)(107,170)
(108,169)(109,176)(110,175)(111,177)(112,173)(113,172)(114,174)(115,180)
(116,179)(117,178)(118,185)(119,184)(120,186)(121,182)(122,181)(123,183)
(124,189)(125,188)(126,187)(191,192)(193,197)(194,196)(195,198)(200,201)
(202,206)(203,205)(204,207)(209,210)(211,215)(212,214)(213,216)(218,219)
(220,224)(221,223)(222,225)(227,228)(229,233)(230,232)(231,234)(236,237)
(238,242)(239,241)(240,243)(245,246)(247,251)(248,250)(249,252)(253,320)
(254,319)(255,321)(256,317)(257,316)(258,318)(259,324)(260,323)(261,322)
(262,329)(263,328)(264,330)(265,326)(266,325)(267,327)(268,333)(269,332)
(270,331)(271,338)(272,337)(273,339)(274,335)(275,334)(276,336)(277,342)
(278,341)(279,340)(280,347)(281,346)(282,348)(283,344)(284,343)(285,345)
(286,351)(287,350)(288,349)(289,356)(290,355)(291,357)(292,353)(293,352)
(294,354)(295,360)(296,359)(297,358)(298,365)(299,364)(300,366)(301,362)
(302,361)(303,363)(304,369)(305,368)(306,367)(307,374)(308,373)(309,375)
(310,371)(311,370)(312,372)(313,378)(314,377)(315,376);;
s1 := (  1, 64)(  2, 66)(  3, 65)(  4, 71)(  5, 70)(  6, 72)(  7, 68)(  8, 67)
(  9, 69)( 10,118)( 11,120)( 12,119)( 13,125)( 14,124)( 15,126)( 16,122)
( 17,121)( 18,123)( 19,109)( 20,111)( 21,110)( 22,116)( 23,115)( 24,117)
( 25,113)( 26,112)( 27,114)( 28,100)( 29,102)( 30,101)( 31,107)( 32,106)
( 33,108)( 34,104)( 35,103)( 36,105)( 37, 91)( 38, 93)( 39, 92)( 40, 98)
( 41, 97)( 42, 99)( 43, 95)( 44, 94)( 45, 96)( 46, 82)( 47, 84)( 48, 83)
( 49, 89)( 50, 88)( 51, 90)( 52, 86)( 53, 85)( 54, 87)( 55, 73)( 56, 75)
( 57, 74)( 58, 80)( 59, 79)( 60, 81)( 61, 77)( 62, 76)( 63, 78)(127,131)
(128,130)(129,132)(133,135)(136,185)(137,184)(138,186)(139,182)(140,181)
(141,183)(142,189)(143,188)(144,187)(145,176)(146,175)(147,177)(148,173)
(149,172)(150,174)(151,180)(152,179)(153,178)(154,167)(155,166)(156,168)
(157,164)(158,163)(159,165)(160,171)(161,170)(162,169)(190,253)(191,255)
(192,254)(193,260)(194,259)(195,261)(196,257)(197,256)(198,258)(199,307)
(200,309)(201,308)(202,314)(203,313)(204,315)(205,311)(206,310)(207,312)
(208,298)(209,300)(210,299)(211,305)(212,304)(213,306)(214,302)(215,301)
(216,303)(217,289)(218,291)(219,290)(220,296)(221,295)(222,297)(223,293)
(224,292)(225,294)(226,280)(227,282)(228,281)(229,287)(230,286)(231,288)
(232,284)(233,283)(234,285)(235,271)(236,273)(237,272)(238,278)(239,277)
(240,279)(241,275)(242,274)(243,276)(244,262)(245,264)(246,263)(247,269)
(248,268)(249,270)(250,266)(251,265)(252,267)(316,320)(317,319)(318,321)
(322,324)(325,374)(326,373)(327,375)(328,371)(329,370)(330,372)(331,378)
(332,377)(333,376)(334,365)(335,364)(336,366)(337,362)(338,361)(339,363)
(340,369)(341,368)(342,367)(343,356)(344,355)(345,357)(346,353)(347,352)
(348,354)(349,360)(350,359)(351,358);;
s2 := (  1,199)(  2,200)(  3,201)(  4,202)(  5,203)(  6,204)(  7,205)(  8,206)
(  9,207)( 10,190)( 11,191)( 12,192)( 13,193)( 14,194)( 15,195)( 16,196)
( 17,197)( 18,198)( 19,244)( 20,245)( 21,246)( 22,247)( 23,248)( 24,249)
( 25,250)( 26,251)( 27,252)( 28,235)( 29,236)( 30,237)( 31,238)( 32,239)
( 33,240)( 34,241)( 35,242)( 36,243)( 37,226)( 38,227)( 39,228)( 40,229)
( 41,230)( 42,231)( 43,232)( 44,233)( 45,234)( 46,217)( 47,218)( 48,219)
( 49,220)( 50,221)( 51,222)( 52,223)( 53,224)( 54,225)( 55,208)( 56,209)
( 57,210)( 58,211)( 59,212)( 60,213)( 61,214)( 62,215)( 63,216)( 64,262)
( 65,263)( 66,264)( 67,265)( 68,266)( 69,267)( 70,268)( 71,269)( 72,270)
( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)( 80,260)
( 81,261)( 82,307)( 83,308)( 84,309)( 85,310)( 86,311)( 87,312)( 88,313)
( 89,314)( 90,315)( 91,298)( 92,299)( 93,300)( 94,301)( 95,302)( 96,303)
( 97,304)( 98,305)( 99,306)(100,289)(101,290)(102,291)(103,292)(104,293)
(105,294)(106,295)(107,296)(108,297)(109,280)(110,281)(111,282)(112,283)
(113,284)(114,285)(115,286)(116,287)(117,288)(118,271)(119,272)(120,273)
(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,325)(128,326)
(129,327)(130,328)(131,329)(132,330)(133,331)(134,332)(135,333)(136,316)
(137,317)(138,318)(139,319)(140,320)(141,321)(142,322)(143,323)(144,324)
(145,370)(146,371)(147,372)(148,373)(149,374)(150,375)(151,376)(152,377)
(153,378)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)(160,367)
(161,368)(162,369)(163,352)(164,353)(165,354)(166,355)(167,356)(168,357)
(169,358)(170,359)(171,360)(172,343)(173,344)(174,345)(175,346)(176,347)
(177,348)(178,349)(179,350)(180,351)(181,334)(182,335)(183,336)(184,337)
(185,338)(186,339)(187,340)(188,341)(189,342);;
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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(378)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)
( 15, 18)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)
( 51, 54)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64,131)( 65,130)( 66,132)
( 67,128)( 68,127)( 69,129)( 70,135)( 71,134)( 72,133)( 73,140)( 74,139)
( 75,141)( 76,137)( 77,136)( 78,138)( 79,144)( 80,143)( 81,142)( 82,149)
( 83,148)( 84,150)( 85,146)( 86,145)( 87,147)( 88,153)( 89,152)( 90,151)
( 91,158)( 92,157)( 93,159)( 94,155)( 95,154)( 96,156)( 97,162)( 98,161)
( 99,160)(100,167)(101,166)(102,168)(103,164)(104,163)(105,165)(106,171)
(107,170)(108,169)(109,176)(110,175)(111,177)(112,173)(113,172)(114,174)
(115,180)(116,179)(117,178)(118,185)(119,184)(120,186)(121,182)(122,181)
(123,183)(124,189)(125,188)(126,187)(191,192)(193,197)(194,196)(195,198)
(200,201)(202,206)(203,205)(204,207)(209,210)(211,215)(212,214)(213,216)
(218,219)(220,224)(221,223)(222,225)(227,228)(229,233)(230,232)(231,234)
(236,237)(238,242)(239,241)(240,243)(245,246)(247,251)(248,250)(249,252)
(253,320)(254,319)(255,321)(256,317)(257,316)(258,318)(259,324)(260,323)
(261,322)(262,329)(263,328)(264,330)(265,326)(266,325)(267,327)(268,333)
(269,332)(270,331)(271,338)(272,337)(273,339)(274,335)(275,334)(276,336)
(277,342)(278,341)(279,340)(280,347)(281,346)(282,348)(283,344)(284,343)
(285,345)(286,351)(287,350)(288,349)(289,356)(290,355)(291,357)(292,353)
(293,352)(294,354)(295,360)(296,359)(297,358)(298,365)(299,364)(300,366)
(301,362)(302,361)(303,363)(304,369)(305,368)(306,367)(307,374)(308,373)
(309,375)(310,371)(311,370)(312,372)(313,378)(314,377)(315,376);
s1 := Sym(378)!(  1, 64)(  2, 66)(  3, 65)(  4, 71)(  5, 70)(  6, 72)(  7, 68)
(  8, 67)(  9, 69)( 10,118)( 11,120)( 12,119)( 13,125)( 14,124)( 15,126)
( 16,122)( 17,121)( 18,123)( 19,109)( 20,111)( 21,110)( 22,116)( 23,115)
( 24,117)( 25,113)( 26,112)( 27,114)( 28,100)( 29,102)( 30,101)( 31,107)
( 32,106)( 33,108)( 34,104)( 35,103)( 36,105)( 37, 91)( 38, 93)( 39, 92)
( 40, 98)( 41, 97)( 42, 99)( 43, 95)( 44, 94)( 45, 96)( 46, 82)( 47, 84)
( 48, 83)( 49, 89)( 50, 88)( 51, 90)( 52, 86)( 53, 85)( 54, 87)( 55, 73)
( 56, 75)( 57, 74)( 58, 80)( 59, 79)( 60, 81)( 61, 77)( 62, 76)( 63, 78)
(127,131)(128,130)(129,132)(133,135)(136,185)(137,184)(138,186)(139,182)
(140,181)(141,183)(142,189)(143,188)(144,187)(145,176)(146,175)(147,177)
(148,173)(149,172)(150,174)(151,180)(152,179)(153,178)(154,167)(155,166)
(156,168)(157,164)(158,163)(159,165)(160,171)(161,170)(162,169)(190,253)
(191,255)(192,254)(193,260)(194,259)(195,261)(196,257)(197,256)(198,258)
(199,307)(200,309)(201,308)(202,314)(203,313)(204,315)(205,311)(206,310)
(207,312)(208,298)(209,300)(210,299)(211,305)(212,304)(213,306)(214,302)
(215,301)(216,303)(217,289)(218,291)(219,290)(220,296)(221,295)(222,297)
(223,293)(224,292)(225,294)(226,280)(227,282)(228,281)(229,287)(230,286)
(231,288)(232,284)(233,283)(234,285)(235,271)(236,273)(237,272)(238,278)
(239,277)(240,279)(241,275)(242,274)(243,276)(244,262)(245,264)(246,263)
(247,269)(248,268)(249,270)(250,266)(251,265)(252,267)(316,320)(317,319)
(318,321)(322,324)(325,374)(326,373)(327,375)(328,371)(329,370)(330,372)
(331,378)(332,377)(333,376)(334,365)(335,364)(336,366)(337,362)(338,361)
(339,363)(340,369)(341,368)(342,367)(343,356)(344,355)(345,357)(346,353)
(347,352)(348,354)(349,360)(350,359)(351,358);
s2 := Sym(378)!(  1,199)(  2,200)(  3,201)(  4,202)(  5,203)(  6,204)(  7,205)
(  8,206)(  9,207)( 10,190)( 11,191)( 12,192)( 13,193)( 14,194)( 15,195)
( 16,196)( 17,197)( 18,198)( 19,244)( 20,245)( 21,246)( 22,247)( 23,248)
( 24,249)( 25,250)( 26,251)( 27,252)( 28,235)( 29,236)( 30,237)( 31,238)
( 32,239)( 33,240)( 34,241)( 35,242)( 36,243)( 37,226)( 38,227)( 39,228)
( 40,229)( 41,230)( 42,231)( 43,232)( 44,233)( 45,234)( 46,217)( 47,218)
( 48,219)( 49,220)( 50,221)( 51,222)( 52,223)( 53,224)( 54,225)( 55,208)
( 56,209)( 57,210)( 58,211)( 59,212)( 60,213)( 61,214)( 62,215)( 63,216)
( 64,262)( 65,263)( 66,264)( 67,265)( 68,266)( 69,267)( 70,268)( 71,269)
( 72,270)( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)
( 80,260)( 81,261)( 82,307)( 83,308)( 84,309)( 85,310)( 86,311)( 87,312)
( 88,313)( 89,314)( 90,315)( 91,298)( 92,299)( 93,300)( 94,301)( 95,302)
( 96,303)( 97,304)( 98,305)( 99,306)(100,289)(101,290)(102,291)(103,292)
(104,293)(105,294)(106,295)(107,296)(108,297)(109,280)(110,281)(111,282)
(112,283)(113,284)(114,285)(115,286)(116,287)(117,288)(118,271)(119,272)
(120,273)(121,274)(122,275)(123,276)(124,277)(125,278)(126,279)(127,325)
(128,326)(129,327)(130,328)(131,329)(132,330)(133,331)(134,332)(135,333)
(136,316)(137,317)(138,318)(139,319)(140,320)(141,321)(142,322)(143,323)
(144,324)(145,370)(146,371)(147,372)(148,373)(149,374)(150,375)(151,376)
(152,377)(153,378)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)
(160,367)(161,368)(162,369)(163,352)(164,353)(165,354)(166,355)(167,356)
(168,357)(169,358)(170,359)(171,360)(172,343)(173,344)(174,345)(175,346)
(176,347)(177,348)(178,349)(179,350)(180,351)(181,334)(182,335)(183,336)
(184,337)(185,338)(186,339)(187,340)(188,341)(189,342);
poly := sub<Sym(378)|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*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 >;

References : None.
to this polytope