# Polytope of Type {390}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {390}*780
Also Known As : 390-gon, {390}. if this polytope has another name.
Group : SmallGroup(780,52)
Rank : 2
Schlafli Type : {390}
Number of vertices, edges, etc : 390, 390
Order of s0s1 : 390
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{390,2} of size 1560
Vertex Figure Of :
{2,390} of size 1560
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {195}*390
3-fold quotients : {130}*260
5-fold quotients : {78}*156
6-fold quotients : {65}*130
10-fold quotients : {39}*78
13-fold quotients : {30}*60
15-fold quotients : {26}*52
26-fold quotients : {15}*30
30-fold quotients : {13}*26
39-fold quotients : {10}*20
65-fold quotients : {6}*12
78-fold quotients : {5}*10
130-fold quotients : {3}*6
195-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {780}*1560
Permutation Representation (GAP) :
```s0 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 53)( 15, 65)
( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 57)
( 24, 56)( 25, 55)( 26, 54)( 27, 40)( 28, 52)( 29, 51)( 30, 50)( 31, 49)
( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 44)( 37, 43)( 38, 42)( 39, 41)
( 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)( 79,183)( 80,195)( 81,194)
( 82,193)( 83,192)( 84,191)( 85,190)( 86,189)( 87,188)( 88,187)( 89,186)
( 90,185)( 91,184)( 92,170)( 93,182)( 94,181)( 95,180)( 96,179)( 97,178)
( 98,177)( 99,176)(100,175)(101,174)(102,173)(103,172)(104,171)(105,157)
(106,169)(107,168)(108,167)(109,166)(110,165)(111,164)(112,163)(113,162)
(114,161)(115,160)(116,159)(117,158)(118,144)(119,156)(120,155)(121,154)
(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)(129,146)
(130,145)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,248)
(210,260)(211,259)(212,258)(213,257)(214,256)(215,255)(216,254)(217,253)
(218,252)(219,251)(220,250)(221,249)(222,235)(223,247)(224,246)(225,245)
(226,244)(227,243)(228,242)(229,241)(230,240)(231,239)(232,238)(233,237)
(234,236)(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)(274,378)(275,390)
(276,389)(277,388)(278,387)(279,386)(280,385)(281,384)(282,383)(283,382)
(284,381)(285,380)(286,379)(287,365)(288,377)(289,376)(290,375)(291,374)
(292,373)(293,372)(294,371)(295,370)(296,369)(297,368)(298,367)(299,366)
(300,352)(301,364)(302,363)(303,362)(304,361)(305,360)(306,359)(307,358)
(308,357)(309,356)(310,355)(311,354)(312,353)(313,339)(314,351)(315,350)
(316,349)(317,348)(318,347)(319,346)(320,345)(321,344)(322,343)(323,342)
(324,341)(325,340);;
s1 := (  1,275)(  2,274)(  3,286)(  4,285)(  5,284)(  6,283)(  7,282)(  8,281)
(  9,280)( 10,279)( 11,278)( 12,277)( 13,276)( 14,262)( 15,261)( 16,273)
( 17,272)( 18,271)( 19,270)( 20,269)( 21,268)( 22,267)( 23,266)( 24,265)
( 25,264)( 26,263)( 27,314)( 28,313)( 29,325)( 30,324)( 31,323)( 32,322)
( 33,321)( 34,320)( 35,319)( 36,318)( 37,317)( 38,316)( 39,315)( 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)( 53,288)( 54,287)( 55,299)( 56,298)
( 57,297)( 58,296)( 59,295)( 60,294)( 61,293)( 62,292)( 63,291)( 64,290)
( 65,289)( 66,210)( 67,209)( 68,221)( 69,220)( 70,219)( 71,218)( 72,217)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,197)( 80,196)
( 81,208)( 82,207)( 83,206)( 84,205)( 85,204)( 86,203)( 87,202)( 88,201)
( 89,200)( 90,199)( 91,198)( 92,249)( 93,248)( 94,260)( 95,259)( 96,258)
( 97,257)( 98,256)( 99,255)(100,254)(101,253)(102,252)(103,251)(104,250)
(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)(118,223)(119,222)(120,234)
(121,233)(122,232)(123,231)(124,230)(125,229)(126,228)(127,227)(128,226)
(129,225)(130,224)(131,340)(132,339)(133,351)(134,350)(135,349)(136,348)
(137,347)(138,346)(139,345)(140,344)(141,343)(142,342)(143,341)(144,327)
(145,326)(146,338)(147,337)(148,336)(149,335)(150,334)(151,333)(152,332)
(153,331)(154,330)(155,329)(156,328)(157,379)(158,378)(159,390)(160,389)
(161,388)(162,387)(163,386)(164,385)(165,384)(166,383)(167,382)(168,381)
(169,380)(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)(183,353)(184,352)
(185,364)(186,363)(187,362)(188,361)(189,360)(190,359)(191,358)(192,357)
(193,356)(194,355)(195,354);;
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*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)( 14, 53)
( 15, 65)( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)
( 23, 57)( 24, 56)( 25, 55)( 26, 54)( 27, 40)( 28, 52)( 29, 51)( 30, 50)
( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)( 36, 44)( 37, 43)( 38, 42)
( 39, 41)( 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)( 79,183)( 80,195)
( 81,194)( 82,193)( 83,192)( 84,191)( 85,190)( 86,189)( 87,188)( 88,187)
( 89,186)( 90,185)( 91,184)( 92,170)( 93,182)( 94,181)( 95,180)( 96,179)
( 97,178)( 98,177)( 99,176)(100,175)(101,174)(102,173)(103,172)(104,171)
(105,157)(106,169)(107,168)(108,167)(109,166)(110,165)(111,164)(112,163)
(113,162)(114,161)(115,160)(116,159)(117,158)(118,144)(119,156)(120,155)
(121,154)(122,153)(123,152)(124,151)(125,150)(126,149)(127,148)(128,147)
(129,146)(130,145)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)
(209,248)(210,260)(211,259)(212,258)(213,257)(214,256)(215,255)(216,254)
(217,253)(218,252)(219,251)(220,250)(221,249)(222,235)(223,247)(224,246)
(225,245)(226,244)(227,243)(228,242)(229,241)(230,240)(231,239)(232,238)
(233,237)(234,236)(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)(274,378)
(275,390)(276,389)(277,388)(278,387)(279,386)(280,385)(281,384)(282,383)
(283,382)(284,381)(285,380)(286,379)(287,365)(288,377)(289,376)(290,375)
(291,374)(292,373)(293,372)(294,371)(295,370)(296,369)(297,368)(298,367)
(299,366)(300,352)(301,364)(302,363)(303,362)(304,361)(305,360)(306,359)
(307,358)(308,357)(309,356)(310,355)(311,354)(312,353)(313,339)(314,351)
(315,350)(316,349)(317,348)(318,347)(319,346)(320,345)(321,344)(322,343)
(323,342)(324,341)(325,340);
s1 := Sym(390)!(  1,275)(  2,274)(  3,286)(  4,285)(  5,284)(  6,283)(  7,282)
(  8,281)(  9,280)( 10,279)( 11,278)( 12,277)( 13,276)( 14,262)( 15,261)
( 16,273)( 17,272)( 18,271)( 19,270)( 20,269)( 21,268)( 22,267)( 23,266)
( 24,265)( 25,264)( 26,263)( 27,314)( 28,313)( 29,325)( 30,324)( 31,323)
( 32,322)( 33,321)( 34,320)( 35,319)( 36,318)( 37,317)( 38,316)( 39,315)
( 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)( 53,288)( 54,287)( 55,299)
( 56,298)( 57,297)( 58,296)( 59,295)( 60,294)( 61,293)( 62,292)( 63,291)
( 64,290)( 65,289)( 66,210)( 67,209)( 68,221)( 69,220)( 70,219)( 71,218)
( 72,217)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,211)( 79,197)
( 80,196)( 81,208)( 82,207)( 83,206)( 84,205)( 85,204)( 86,203)( 87,202)
( 88,201)( 89,200)( 90,199)( 91,198)( 92,249)( 93,248)( 94,260)( 95,259)
( 96,258)( 97,257)( 98,256)( 99,255)(100,254)(101,253)(102,252)(103,251)
(104,250)(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)(118,223)(119,222)
(120,234)(121,233)(122,232)(123,231)(124,230)(125,229)(126,228)(127,227)
(128,226)(129,225)(130,224)(131,340)(132,339)(133,351)(134,350)(135,349)
(136,348)(137,347)(138,346)(139,345)(140,344)(141,343)(142,342)(143,341)
(144,327)(145,326)(146,338)(147,337)(148,336)(149,335)(150,334)(151,333)
(152,332)(153,331)(154,330)(155,329)(156,328)(157,379)(158,378)(159,390)
(160,389)(161,388)(162,387)(163,386)(164,385)(165,384)(166,383)(167,382)
(168,381)(169,380)(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)(183,353)
(184,352)(185,364)(186,363)(187,362)(188,361)(189,360)(190,359)(191,358)
(192,357)(193,356)(194,355)(195,354);
poly := sub<Sym(390)|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*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