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Polytope of Type {12,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,30}*1440a
if this polytope has a name.
Group : SmallGroup(1440,5900)
Rank : 3
Schlafli Type : {12,30}
Number of vertices, edges, etc : 24, 360, 60
Order of s0s1s2 : 30
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,30}*720d
3-fold quotients : {4,30}*480
4-fold quotients : {6,30}*360b
5-fold quotients : {12,6}*288a
6-fold quotients : {4,15}*240, {4,30}*240b, {4,30}*240c
10-fold quotients : {12,6}*144d
12-fold quotients : {4,15}*120, {6,10}*120, {2,30}*120
15-fold quotients : {4,6}*96
20-fold quotients : {6,6}*72a
24-fold quotients : {2,15}*60
30-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
36-fold quotients : {2,10}*40
60-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
72-fold quotients : {2,5}*20
120-fold quotients : {2,3}*12, {3,2}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 43)( 22, 44)( 23, 41)( 24, 42)( 25, 47)( 26, 48)
( 27, 45)( 28, 46)( 29, 51)( 30, 52)( 31, 49)( 32, 50)( 33, 55)( 34, 56)
( 35, 53)( 36, 54)( 37, 59)( 38, 60)( 39, 57)( 40, 58)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81,103)( 82,104)( 83,101)( 84,102)( 85,107)( 86,108)( 87,105)( 88,106)
( 89,111)( 90,112)( 91,109)( 92,110)( 93,115)( 94,116)( 95,113)( 96,114)
( 97,119)( 98,120)( 99,117)(100,118)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,163)(142,164)
(143,161)(144,162)(145,167)(146,168)(147,165)(148,166)(149,171)(150,172)
(151,169)(152,170)(153,175)(154,176)(155,173)(156,174)(157,179)(158,180)
(159,177)(160,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,223)(202,224)(203,221)(204,222)
(205,227)(206,228)(207,225)(208,226)(209,231)(210,232)(211,229)(212,230)
(213,235)(214,236)(215,233)(216,234)(217,239)(218,240)(219,237)(220,238)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,283)(262,284)(263,281)(264,282)(265,287)(266,288)
(267,285)(268,286)(269,291)(270,292)(271,289)(272,290)(273,295)(274,296)
(275,293)(276,294)(277,299)(278,300)(279,297)(280,298)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,343)(322,344)(323,341)(324,342)(325,347)(326,348)(327,345)(328,346)
(329,351)(330,352)(331,349)(332,350)(333,355)(334,356)(335,353)(336,354)
(337,359)(338,360)(339,357)(340,358);;
s1 := (  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 37)(  6, 38)(  7, 40)(  8, 39)
(  9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)( 16, 31)
( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)( 47, 60)
( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61,141)( 62,142)( 63,144)
( 64,143)( 65,157)( 66,158)( 67,160)( 68,159)( 69,153)( 70,154)( 71,156)
( 72,155)( 73,149)( 74,150)( 75,152)( 76,151)( 77,145)( 78,146)( 79,148)
( 80,147)( 81,121)( 82,122)( 83,124)( 84,123)( 85,137)( 86,138)( 87,140)
( 88,139)( 89,133)( 90,134)( 91,136)( 92,135)( 93,129)( 94,130)( 95,132)
( 96,131)( 97,125)( 98,126)( 99,128)(100,127)(101,161)(102,162)(103,164)
(104,163)(105,177)(106,178)(107,180)(108,179)(109,173)(110,174)(111,176)
(112,175)(113,169)(114,170)(115,172)(116,171)(117,165)(118,166)(119,168)
(120,167)(181,201)(182,202)(183,204)(184,203)(185,217)(186,218)(187,220)
(188,219)(189,213)(190,214)(191,216)(192,215)(193,209)(194,210)(195,212)
(196,211)(197,205)(198,206)(199,208)(200,207)(223,224)(225,237)(226,238)
(227,240)(228,239)(229,233)(230,234)(231,236)(232,235)(241,321)(242,322)
(243,324)(244,323)(245,337)(246,338)(247,340)(248,339)(249,333)(250,334)
(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,325)(258,326)
(259,328)(260,327)(261,301)(262,302)(263,304)(264,303)(265,317)(266,318)
(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,309)(274,310)
(275,312)(276,311)(277,305)(278,306)(279,308)(280,307)(281,341)(282,342)
(283,344)(284,343)(285,357)(286,358)(287,360)(288,359)(289,353)(290,354)
(291,356)(292,355)(293,349)(294,350)(295,352)(296,351)(297,345)(298,346)
(299,348)(300,347);;
s2 := (  1,245)(  2,248)(  3,247)(  4,246)(  5,241)(  6,244)(  7,243)(  8,242)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,249)( 18,252)( 19,251)( 20,250)( 21,265)( 22,268)( 23,267)( 24,266)
( 25,261)( 26,264)( 27,263)( 28,262)( 29,277)( 30,280)( 31,279)( 32,278)
( 33,273)( 34,276)( 35,275)( 36,274)( 37,269)( 38,272)( 39,271)( 40,270)
( 41,285)( 42,288)( 43,287)( 44,286)( 45,281)( 46,284)( 47,283)( 48,282)
( 49,297)( 50,300)( 51,299)( 52,298)( 53,293)( 54,296)( 55,295)( 56,294)
( 57,289)( 58,292)( 59,291)( 60,290)( 61,185)( 62,188)( 63,187)( 64,186)
( 65,181)( 66,184)( 67,183)( 68,182)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)( 80,190)
( 81,205)( 82,208)( 83,207)( 84,206)( 85,201)( 86,204)( 87,203)( 88,202)
( 89,217)( 90,220)( 91,219)( 92,218)( 93,213)( 94,216)( 95,215)( 96,214)
( 97,209)( 98,212)( 99,211)(100,210)(101,225)(102,228)(103,227)(104,226)
(105,221)(106,224)(107,223)(108,222)(109,237)(110,240)(111,239)(112,238)
(113,233)(114,236)(115,235)(116,234)(117,229)(118,232)(119,231)(120,230)
(121,305)(122,308)(123,307)(124,306)(125,301)(126,304)(127,303)(128,302)
(129,317)(130,320)(131,319)(132,318)(133,313)(134,316)(135,315)(136,314)
(137,309)(138,312)(139,311)(140,310)(141,325)(142,328)(143,327)(144,326)
(145,321)(146,324)(147,323)(148,322)(149,337)(150,340)(151,339)(152,338)
(153,333)(154,336)(155,335)(156,334)(157,329)(158,332)(159,331)(160,330)
(161,345)(162,348)(163,347)(164,346)(165,341)(166,344)(167,343)(168,342)
(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)
(177,349)(178,352)(179,351)(180,350);;
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*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*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(360)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 43)( 22, 44)( 23, 41)( 24, 42)( 25, 47)
( 26, 48)( 27, 45)( 28, 46)( 29, 51)( 30, 52)( 31, 49)( 32, 50)( 33, 55)
( 34, 56)( 35, 53)( 36, 54)( 37, 59)( 38, 60)( 39, 57)( 40, 58)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81,103)( 82,104)( 83,101)( 84,102)( 85,107)( 86,108)( 87,105)
( 88,106)( 89,111)( 90,112)( 91,109)( 92,110)( 93,115)( 94,116)( 95,113)
( 96,114)( 97,119)( 98,120)( 99,117)(100,118)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,163)
(142,164)(143,161)(144,162)(145,167)(146,168)(147,165)(148,166)(149,171)
(150,172)(151,169)(152,170)(153,175)(154,176)(155,173)(156,174)(157,179)
(158,180)(159,177)(160,178)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,223)(202,224)(203,221)
(204,222)(205,227)(206,228)(207,225)(208,226)(209,231)(210,232)(211,229)
(212,230)(213,235)(214,236)(215,233)(216,234)(217,239)(218,240)(219,237)
(220,238)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,283)(262,284)(263,281)(264,282)(265,287)
(266,288)(267,285)(268,286)(269,291)(270,292)(271,289)(272,290)(273,295)
(274,296)(275,293)(276,294)(277,299)(278,300)(279,297)(280,298)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,343)(322,344)(323,341)(324,342)(325,347)(326,348)(327,345)
(328,346)(329,351)(330,352)(331,349)(332,350)(333,355)(334,356)(335,353)
(336,354)(337,359)(338,360)(339,357)(340,358);
s1 := Sym(360)!(  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 37)(  6, 38)(  7, 40)
(  8, 39)(  9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 17, 25)( 18, 26)( 19, 28)( 20, 27)( 43, 44)( 45, 57)( 46, 58)
( 47, 60)( 48, 59)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 61,141)( 62,142)
( 63,144)( 64,143)( 65,157)( 66,158)( 67,160)( 68,159)( 69,153)( 70,154)
( 71,156)( 72,155)( 73,149)( 74,150)( 75,152)( 76,151)( 77,145)( 78,146)
( 79,148)( 80,147)( 81,121)( 82,122)( 83,124)( 84,123)( 85,137)( 86,138)
( 87,140)( 88,139)( 89,133)( 90,134)( 91,136)( 92,135)( 93,129)( 94,130)
( 95,132)( 96,131)( 97,125)( 98,126)( 99,128)(100,127)(101,161)(102,162)
(103,164)(104,163)(105,177)(106,178)(107,180)(108,179)(109,173)(110,174)
(111,176)(112,175)(113,169)(114,170)(115,172)(116,171)(117,165)(118,166)
(119,168)(120,167)(181,201)(182,202)(183,204)(184,203)(185,217)(186,218)
(187,220)(188,219)(189,213)(190,214)(191,216)(192,215)(193,209)(194,210)
(195,212)(196,211)(197,205)(198,206)(199,208)(200,207)(223,224)(225,237)
(226,238)(227,240)(228,239)(229,233)(230,234)(231,236)(232,235)(241,321)
(242,322)(243,324)(244,323)(245,337)(246,338)(247,340)(248,339)(249,333)
(250,334)(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,325)
(258,326)(259,328)(260,327)(261,301)(262,302)(263,304)(264,303)(265,317)
(266,318)(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,309)
(274,310)(275,312)(276,311)(277,305)(278,306)(279,308)(280,307)(281,341)
(282,342)(283,344)(284,343)(285,357)(286,358)(287,360)(288,359)(289,353)
(290,354)(291,356)(292,355)(293,349)(294,350)(295,352)(296,351)(297,345)
(298,346)(299,348)(300,347);
s2 := Sym(360)!(  1,245)(  2,248)(  3,247)(  4,246)(  5,241)(  6,244)(  7,243)
(  8,242)(  9,257)( 10,260)( 11,259)( 12,258)( 13,253)( 14,256)( 15,255)
( 16,254)( 17,249)( 18,252)( 19,251)( 20,250)( 21,265)( 22,268)( 23,267)
( 24,266)( 25,261)( 26,264)( 27,263)( 28,262)( 29,277)( 30,280)( 31,279)
( 32,278)( 33,273)( 34,276)( 35,275)( 36,274)( 37,269)( 38,272)( 39,271)
( 40,270)( 41,285)( 42,288)( 43,287)( 44,286)( 45,281)( 46,284)( 47,283)
( 48,282)( 49,297)( 50,300)( 51,299)( 52,298)( 53,293)( 54,296)( 55,295)
( 56,294)( 57,289)( 58,292)( 59,291)( 60,290)( 61,185)( 62,188)( 63,187)
( 64,186)( 65,181)( 66,184)( 67,183)( 68,182)( 69,197)( 70,200)( 71,199)
( 72,198)( 73,193)( 74,196)( 75,195)( 76,194)( 77,189)( 78,192)( 79,191)
( 80,190)( 81,205)( 82,208)( 83,207)( 84,206)( 85,201)( 86,204)( 87,203)
( 88,202)( 89,217)( 90,220)( 91,219)( 92,218)( 93,213)( 94,216)( 95,215)
( 96,214)( 97,209)( 98,212)( 99,211)(100,210)(101,225)(102,228)(103,227)
(104,226)(105,221)(106,224)(107,223)(108,222)(109,237)(110,240)(111,239)
(112,238)(113,233)(114,236)(115,235)(116,234)(117,229)(118,232)(119,231)
(120,230)(121,305)(122,308)(123,307)(124,306)(125,301)(126,304)(127,303)
(128,302)(129,317)(130,320)(131,319)(132,318)(133,313)(134,316)(135,315)
(136,314)(137,309)(138,312)(139,311)(140,310)(141,325)(142,328)(143,327)
(144,326)(145,321)(146,324)(147,323)(148,322)(149,337)(150,340)(151,339)
(152,338)(153,333)(154,336)(155,335)(156,334)(157,329)(158,332)(159,331)
(160,330)(161,345)(162,348)(163,347)(164,346)(165,341)(166,344)(167,343)
(168,342)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)
(176,354)(177,349)(178,352)(179,351)(180,350);
poly := sub<Sym(360)|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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*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*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