Polytope of Type {378}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {378}*756
Also Known As : 378-gon, {378}. if this polytope has another name.
Group : SmallGroup(756,14)
Rank : 2
Schlafli Type : {378}
Number of vertices, edges, etc : 378, 378
Order of s0s1 : 378
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{378,2} of size 1512
Vertex Figure Of :
{2,378} of size 1512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {189}*378
3-fold quotients : {126}*252
6-fold quotients : {63}*126
7-fold quotients : {54}*108
9-fold quotients : {42}*84
14-fold quotients : {27}*54
18-fold quotients : {21}*42
21-fold quotients : {18}*36
27-fold quotients : {14}*28
42-fold quotients : {9}*18
54-fold quotients : {7}*14
63-fold quotients : {6}*12
126-fold quotients : {3}*6
189-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {756}*1512
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 55)( 11, 57)( 12, 56)( 13, 63)
( 14, 62)( 15, 61)( 16, 60)( 17, 59)( 18, 58)( 19, 46)( 20, 48)( 21, 47)
( 22, 54)( 23, 53)( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 28, 37)( 29, 39)
( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 64,135)
( 65,134)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,128)( 72,127)
( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,184)( 79,183)( 80,182)
( 81,181)( 82,180)( 83,179)( 84,178)( 85,177)( 86,176)( 87,175)( 88,174)
( 89,173)( 90,172)( 91,171)( 92,170)( 93,169)( 94,168)( 95,167)( 96,166)
( 97,165)( 98,164)( 99,163)(100,162)(101,161)(102,160)(103,159)(104,158)
(105,157)(106,156)(107,155)(108,154)(109,153)(110,152)(111,151)(112,150)
(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)
(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(191,192)(193,198)
(194,197)(195,196)(199,244)(200,246)(201,245)(202,252)(203,251)(204,250)
(205,249)(206,248)(207,247)(208,235)(209,237)(210,236)(211,243)(212,242)
(213,241)(214,240)(215,239)(216,238)(217,226)(218,228)(219,227)(220,234)
(221,233)(222,232)(223,231)(224,230)(225,229)(253,324)(254,323)(255,322)
(256,321)(257,320)(258,319)(259,318)(260,317)(261,316)(262,378)(263,377)
(264,376)(265,375)(266,374)(267,373)(268,372)(269,371)(270,370)(271,369)
(272,368)(273,367)(274,366)(275,365)(276,364)(277,363)(278,362)(279,361)
(280,360)(281,359)(282,358)(283,357)(284,356)(285,355)(286,354)(287,353)
(288,352)(289,351)(290,350)(291,349)(292,348)(293,347)(294,346)(295,345)
(296,344)(297,343)(298,342)(299,341)(300,340)(301,339)(302,338)(303,337)
(304,336)(305,335)(306,334)(307,333)(308,332)(309,331)(310,330)(311,329)
(312,328)(313,327)(314,326)(315,325);;
s1 := ( 1,262)( 2,264)( 3,263)( 4,270)( 5,269)( 6,268)( 7,267)( 8,266)
( 9,265)( 10,253)( 11,255)( 12,254)( 13,261)( 14,260)( 15,259)( 16,258)
( 17,257)( 18,256)( 19,307)( 20,309)( 21,308)( 22,315)( 23,314)( 24,313)
( 25,312)( 26,311)( 27,310)( 28,298)( 29,300)( 30,299)( 31,306)( 32,305)
( 33,304)( 34,303)( 35,302)( 36,301)( 37,289)( 38,291)( 39,290)( 40,297)
( 41,296)( 42,295)( 43,294)( 44,293)( 45,292)( 46,280)( 47,282)( 48,281)
( 49,288)( 50,287)( 51,286)( 52,285)( 53,284)( 54,283)( 55,271)( 56,273)
( 57,272)( 58,279)( 59,278)( 60,277)( 61,276)( 62,275)( 63,274)( 64,199)
( 65,201)( 66,200)( 67,207)( 68,206)( 69,205)( 70,204)( 71,203)( 72,202)
( 73,190)( 74,192)( 75,191)( 76,198)( 77,197)( 78,196)( 79,195)( 80,194)
( 81,193)( 82,244)( 83,246)( 84,245)( 85,252)( 86,251)( 87,250)( 88,249)
( 89,248)( 90,247)( 91,235)( 92,237)( 93,236)( 94,243)( 95,242)( 96,241)
( 97,240)( 98,239)( 99,238)(100,226)(101,228)(102,227)(103,234)(104,233)
(105,232)(106,231)(107,230)(108,229)(109,217)(110,219)(111,218)(112,225)
(113,224)(114,223)(115,222)(116,221)(117,220)(118,208)(119,210)(120,209)
(121,216)(122,215)(123,214)(124,213)(125,212)(126,211)(127,333)(128,332)
(129,331)(130,330)(131,329)(132,328)(133,327)(134,326)(135,325)(136,324)
(137,323)(138,322)(139,321)(140,320)(141,319)(142,318)(143,317)(144,316)
(145,378)(146,377)(147,376)(148,375)(149,374)(150,373)(151,372)(152,371)
(153,370)(154,369)(155,368)(156,367)(157,366)(158,365)(159,364)(160,363)
(161,362)(162,361)(163,360)(164,359)(165,358)(166,357)(167,356)(168,355)
(169,354)(170,353)(171,352)(172,351)(173,350)(174,349)(175,348)(176,347)
(177,346)(178,345)(179,344)(180,343)(181,342)(182,341)(183,340)(184,339)
(185,338)(186,337)(187,336)(188,335)(189,334);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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, 9)( 5, 8)( 6, 7)( 10, 55)( 11, 57)( 12, 56)
( 13, 63)( 14, 62)( 15, 61)( 16, 60)( 17, 59)( 18, 58)( 19, 46)( 20, 48)
( 21, 47)( 22, 54)( 23, 53)( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 28, 37)
( 29, 39)( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)
( 64,135)( 65,134)( 66,133)( 67,132)( 68,131)( 69,130)( 70,129)( 71,128)
( 72,127)( 73,189)( 74,188)( 75,187)( 76,186)( 77,185)( 78,184)( 79,183)
( 80,182)( 81,181)( 82,180)( 83,179)( 84,178)( 85,177)( 86,176)( 87,175)
( 88,174)( 89,173)( 90,172)( 91,171)( 92,170)( 93,169)( 94,168)( 95,167)
( 96,166)( 97,165)( 98,164)( 99,163)(100,162)(101,161)(102,160)(103,159)
(104,158)(105,157)(106,156)(107,155)(108,154)(109,153)(110,152)(111,151)
(112,150)(113,149)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)
(120,142)(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(191,192)
(193,198)(194,197)(195,196)(199,244)(200,246)(201,245)(202,252)(203,251)
(204,250)(205,249)(206,248)(207,247)(208,235)(209,237)(210,236)(211,243)
(212,242)(213,241)(214,240)(215,239)(216,238)(217,226)(218,228)(219,227)
(220,234)(221,233)(222,232)(223,231)(224,230)(225,229)(253,324)(254,323)
(255,322)(256,321)(257,320)(258,319)(259,318)(260,317)(261,316)(262,378)
(263,377)(264,376)(265,375)(266,374)(267,373)(268,372)(269,371)(270,370)
(271,369)(272,368)(273,367)(274,366)(275,365)(276,364)(277,363)(278,362)
(279,361)(280,360)(281,359)(282,358)(283,357)(284,356)(285,355)(286,354)
(287,353)(288,352)(289,351)(290,350)(291,349)(292,348)(293,347)(294,346)
(295,345)(296,344)(297,343)(298,342)(299,341)(300,340)(301,339)(302,338)
(303,337)(304,336)(305,335)(306,334)(307,333)(308,332)(309,331)(310,330)
(311,329)(312,328)(313,327)(314,326)(315,325);
s1 := Sym(378)!( 1,262)( 2,264)( 3,263)( 4,270)( 5,269)( 6,268)( 7,267)
( 8,266)( 9,265)( 10,253)( 11,255)( 12,254)( 13,261)( 14,260)( 15,259)
( 16,258)( 17,257)( 18,256)( 19,307)( 20,309)( 21,308)( 22,315)( 23,314)
( 24,313)( 25,312)( 26,311)( 27,310)( 28,298)( 29,300)( 30,299)( 31,306)
( 32,305)( 33,304)( 34,303)( 35,302)( 36,301)( 37,289)( 38,291)( 39,290)
( 40,297)( 41,296)( 42,295)( 43,294)( 44,293)( 45,292)( 46,280)( 47,282)
( 48,281)( 49,288)( 50,287)( 51,286)( 52,285)( 53,284)( 54,283)( 55,271)
( 56,273)( 57,272)( 58,279)( 59,278)( 60,277)( 61,276)( 62,275)( 63,274)
( 64,199)( 65,201)( 66,200)( 67,207)( 68,206)( 69,205)( 70,204)( 71,203)
( 72,202)( 73,190)( 74,192)( 75,191)( 76,198)( 77,197)( 78,196)( 79,195)
( 80,194)( 81,193)( 82,244)( 83,246)( 84,245)( 85,252)( 86,251)( 87,250)
( 88,249)( 89,248)( 90,247)( 91,235)( 92,237)( 93,236)( 94,243)( 95,242)
( 96,241)( 97,240)( 98,239)( 99,238)(100,226)(101,228)(102,227)(103,234)
(104,233)(105,232)(106,231)(107,230)(108,229)(109,217)(110,219)(111,218)
(112,225)(113,224)(114,223)(115,222)(116,221)(117,220)(118,208)(119,210)
(120,209)(121,216)(122,215)(123,214)(124,213)(125,212)(126,211)(127,333)
(128,332)(129,331)(130,330)(131,329)(132,328)(133,327)(134,326)(135,325)
(136,324)(137,323)(138,322)(139,321)(140,320)(141,319)(142,318)(143,317)
(144,316)(145,378)(146,377)(147,376)(148,375)(149,374)(150,373)(151,372)
(152,371)(153,370)(154,369)(155,368)(156,367)(157,366)(158,365)(159,364)
(160,363)(161,362)(162,361)(163,360)(164,359)(165,358)(166,357)(167,356)
(168,355)(169,354)(170,353)(171,352)(172,351)(173,350)(174,349)(175,348)
(176,347)(177,346)(178,345)(179,344)(180,343)(181,342)(182,341)(183,340)
(184,339)(185,338)(186,337)(187,336)(188,335)(189,334);
poly := sub<Sym(378)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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