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# Polytope of Type {6,30}

Atlas Canonical Name : {6,30}*1440h
if this polytope has a name.
Group : SmallGroup(1440,5901)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 24, 360, 120
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
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 : {6,15}*720e
3-fold quotients : {6,30}*480
4-fold quotients : {6,30}*360c
5-fold quotients : {6,6}*288a
6-fold quotients : {6,15}*240
8-fold quotients : {6,15}*180
10-fold quotients : {6,3}*144
12-fold quotients : {2,30}*120
15-fold quotients : {6,6}*96
20-fold quotients : {6,6}*72b
24-fold quotients : {2,15}*60
30-fold quotients : {3,6}*48, {6,3}*48
36-fold quotients : {2,10}*40
40-fold quotients : {6,3}*36
60-fold quotients : {3,3}*24, {2,6}*24
72-fold quotients : {2,5}*20
120-fold quotients : {2,3}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 61,121)
( 62,122)( 63,124)( 64,123)( 65,125)( 66,126)( 67,128)( 68,127)( 69,129)
( 70,130)( 71,132)( 72,131)( 73,133)( 74,134)( 75,136)( 76,135)( 77,137)
( 78,138)( 79,140)( 80,139)( 81,141)( 82,142)( 83,144)( 84,143)( 85,145)
( 86,146)( 87,148)( 88,147)( 89,149)( 90,150)( 91,152)( 92,151)( 93,153)
( 94,154)( 95,156)( 96,155)( 97,157)( 98,158)( 99,160)(100,159)(101,161)
(102,162)(103,164)(104,163)(105,165)(106,166)(107,168)(108,167)(109,169)
(110,170)(111,172)(112,171)(113,173)(114,174)(115,176)(116,175)(117,177)
(118,178)(119,180)(120,179)(183,184)(187,188)(191,192)(195,196)(199,200)
(203,204)(207,208)(211,212)(215,216)(219,220)(223,224)(227,228)(231,232)
(235,236)(239,240)(241,301)(242,302)(243,304)(244,303)(245,305)(246,306)
(247,308)(248,307)(249,309)(250,310)(251,312)(252,311)(253,313)(254,314)
(255,316)(256,315)(257,317)(258,318)(259,320)(260,319)(261,321)(262,322)
(263,324)(264,323)(265,325)(266,326)(267,328)(268,327)(269,329)(270,330)
(271,332)(272,331)(273,333)(274,334)(275,336)(276,335)(277,337)(278,338)
(279,340)(280,339)(281,341)(282,342)(283,344)(284,343)(285,345)(286,346)
(287,348)(288,347)(289,349)(290,350)(291,352)(292,351)(293,353)(294,354)
(295,356)(296,355)(297,357)(298,358)(299,360)(300,359);;
s1 := (  1, 61)(  2, 64)(  3, 63)(  4, 62)(  5, 77)(  6, 80)(  7, 79)(  8, 78)
(  9, 73)( 10, 76)( 11, 75)( 12, 74)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21,101)( 22,104)( 23,103)( 24,102)
( 25,117)( 26,120)( 27,119)( 28,118)( 29,113)( 30,116)( 31,115)( 32,114)
( 33,109)( 34,112)( 35,111)( 36,110)( 37,105)( 38,108)( 39,107)( 40,106)
( 41, 81)( 42, 84)( 43, 83)( 44, 82)( 45, 97)( 46,100)( 47, 99)( 48, 98)
( 49, 93)( 50, 96)( 51, 95)( 52, 94)( 53, 89)( 54, 92)( 55, 91)( 56, 90)
( 57, 85)( 58, 88)( 59, 87)( 60, 86)(122,124)(125,137)(126,140)(127,139)
(128,138)(129,133)(130,136)(131,135)(132,134)(141,161)(142,164)(143,163)
(144,162)(145,177)(146,180)(147,179)(148,178)(149,173)(150,176)(151,175)
(152,174)(153,169)(154,172)(155,171)(156,170)(157,165)(158,168)(159,167)
(160,166)(181,241)(182,244)(183,243)(184,242)(185,257)(186,260)(187,259)
(188,258)(189,253)(190,256)(191,255)(192,254)(193,249)(194,252)(195,251)
(196,250)(197,245)(198,248)(199,247)(200,246)(201,281)(202,284)(203,283)
(204,282)(205,297)(206,300)(207,299)(208,298)(209,293)(210,296)(211,295)
(212,294)(213,289)(214,292)(215,291)(216,290)(217,285)(218,288)(219,287)
(220,286)(221,261)(222,264)(223,263)(224,262)(225,277)(226,280)(227,279)
(228,278)(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)(235,271)
(236,270)(237,265)(238,268)(239,267)(240,266)(302,304)(305,317)(306,320)
(307,319)(308,318)(309,313)(310,316)(311,315)(312,314)(321,341)(322,344)
(323,343)(324,342)(325,357)(326,360)(327,359)(328,358)(329,353)(330,356)
(331,355)(332,354)(333,349)(334,352)(335,351)(336,350)(337,345)(338,348)
(339,347)(340,346);;
s2 := (  1,206)(  2,205)(  3,207)(  4,208)(  5,202)(  6,201)(  7,203)(  8,204)
(  9,218)( 10,217)( 11,219)( 12,220)( 13,214)( 14,213)( 15,215)( 16,216)
( 17,210)( 18,209)( 19,211)( 20,212)( 21,186)( 22,185)( 23,187)( 24,188)
( 25,182)( 26,181)( 27,183)( 28,184)( 29,198)( 30,197)( 31,199)( 32,200)
( 33,194)( 34,193)( 35,195)( 36,196)( 37,190)( 38,189)( 39,191)( 40,192)
( 41,226)( 42,225)( 43,227)( 44,228)( 45,222)( 46,221)( 47,223)( 48,224)
( 49,238)( 50,237)( 51,239)( 52,240)( 53,234)( 54,233)( 55,235)( 56,236)
( 57,230)( 58,229)( 59,231)( 60,232)( 61,326)( 62,325)( 63,327)( 64,328)
( 65,322)( 66,321)( 67,323)( 68,324)( 69,338)( 70,337)( 71,339)( 72,340)
( 73,334)( 74,333)( 75,335)( 76,336)( 77,330)( 78,329)( 79,331)( 80,332)
( 81,306)( 82,305)( 83,307)( 84,308)( 85,302)( 86,301)( 87,303)( 88,304)
( 89,318)( 90,317)( 91,319)( 92,320)( 93,314)( 94,313)( 95,315)( 96,316)
( 97,310)( 98,309)( 99,311)(100,312)(101,346)(102,345)(103,347)(104,348)
(105,342)(106,341)(107,343)(108,344)(109,358)(110,357)(111,359)(112,360)
(113,354)(114,353)(115,355)(116,356)(117,350)(118,349)(119,351)(120,352)
(121,266)(122,265)(123,267)(124,268)(125,262)(126,261)(127,263)(128,264)
(129,278)(130,277)(131,279)(132,280)(133,274)(134,273)(135,275)(136,276)
(137,270)(138,269)(139,271)(140,272)(141,246)(142,245)(143,247)(144,248)
(145,242)(146,241)(147,243)(148,244)(149,258)(150,257)(151,259)(152,260)
(153,254)(154,253)(155,255)(156,256)(157,250)(158,249)(159,251)(160,252)
(161,286)(162,285)(163,287)(164,288)(165,282)(166,281)(167,283)(168,284)
(169,298)(170,297)(171,299)(172,300)(173,294)(174,293)(175,295)(176,296)
(177,290)(178,289)(179,291)(180,292);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*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)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)
( 61,121)( 62,122)( 63,124)( 64,123)( 65,125)( 66,126)( 67,128)( 68,127)
( 69,129)( 70,130)( 71,132)( 72,131)( 73,133)( 74,134)( 75,136)( 76,135)
( 77,137)( 78,138)( 79,140)( 80,139)( 81,141)( 82,142)( 83,144)( 84,143)
( 85,145)( 86,146)( 87,148)( 88,147)( 89,149)( 90,150)( 91,152)( 92,151)
( 93,153)( 94,154)( 95,156)( 96,155)( 97,157)( 98,158)( 99,160)(100,159)
(101,161)(102,162)(103,164)(104,163)(105,165)(106,166)(107,168)(108,167)
(109,169)(110,170)(111,172)(112,171)(113,173)(114,174)(115,176)(116,175)
(117,177)(118,178)(119,180)(120,179)(183,184)(187,188)(191,192)(195,196)
(199,200)(203,204)(207,208)(211,212)(215,216)(219,220)(223,224)(227,228)
(231,232)(235,236)(239,240)(241,301)(242,302)(243,304)(244,303)(245,305)
(246,306)(247,308)(248,307)(249,309)(250,310)(251,312)(252,311)(253,313)
(254,314)(255,316)(256,315)(257,317)(258,318)(259,320)(260,319)(261,321)
(262,322)(263,324)(264,323)(265,325)(266,326)(267,328)(268,327)(269,329)
(270,330)(271,332)(272,331)(273,333)(274,334)(275,336)(276,335)(277,337)
(278,338)(279,340)(280,339)(281,341)(282,342)(283,344)(284,343)(285,345)
(286,346)(287,348)(288,347)(289,349)(290,350)(291,352)(292,351)(293,353)
(294,354)(295,356)(296,355)(297,357)(298,358)(299,360)(300,359);
s1 := Sym(360)!(  1, 61)(  2, 64)(  3, 63)(  4, 62)(  5, 77)(  6, 80)(  7, 79)
(  8, 78)(  9, 73)( 10, 76)( 11, 75)( 12, 74)( 13, 69)( 14, 72)( 15, 71)
( 16, 70)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21,101)( 22,104)( 23,103)
( 24,102)( 25,117)( 26,120)( 27,119)( 28,118)( 29,113)( 30,116)( 31,115)
( 32,114)( 33,109)( 34,112)( 35,111)( 36,110)( 37,105)( 38,108)( 39,107)
( 40,106)( 41, 81)( 42, 84)( 43, 83)( 44, 82)( 45, 97)( 46,100)( 47, 99)
( 48, 98)( 49, 93)( 50, 96)( 51, 95)( 52, 94)( 53, 89)( 54, 92)( 55, 91)
( 56, 90)( 57, 85)( 58, 88)( 59, 87)( 60, 86)(122,124)(125,137)(126,140)
(127,139)(128,138)(129,133)(130,136)(131,135)(132,134)(141,161)(142,164)
(143,163)(144,162)(145,177)(146,180)(147,179)(148,178)(149,173)(150,176)
(151,175)(152,174)(153,169)(154,172)(155,171)(156,170)(157,165)(158,168)
(159,167)(160,166)(181,241)(182,244)(183,243)(184,242)(185,257)(186,260)
(187,259)(188,258)(189,253)(190,256)(191,255)(192,254)(193,249)(194,252)
(195,251)(196,250)(197,245)(198,248)(199,247)(200,246)(201,281)(202,284)
(203,283)(204,282)(205,297)(206,300)(207,299)(208,298)(209,293)(210,296)
(211,295)(212,294)(213,289)(214,292)(215,291)(216,290)(217,285)(218,288)
(219,287)(220,286)(221,261)(222,264)(223,263)(224,262)(225,277)(226,280)
(227,279)(228,278)(229,273)(230,276)(231,275)(232,274)(233,269)(234,272)
(235,271)(236,270)(237,265)(238,268)(239,267)(240,266)(302,304)(305,317)
(306,320)(307,319)(308,318)(309,313)(310,316)(311,315)(312,314)(321,341)
(322,344)(323,343)(324,342)(325,357)(326,360)(327,359)(328,358)(329,353)
(330,356)(331,355)(332,354)(333,349)(334,352)(335,351)(336,350)(337,345)
(338,348)(339,347)(340,346);
s2 := Sym(360)!(  1,206)(  2,205)(  3,207)(  4,208)(  5,202)(  6,201)(  7,203)
(  8,204)(  9,218)( 10,217)( 11,219)( 12,220)( 13,214)( 14,213)( 15,215)
( 16,216)( 17,210)( 18,209)( 19,211)( 20,212)( 21,186)( 22,185)( 23,187)
( 24,188)( 25,182)( 26,181)( 27,183)( 28,184)( 29,198)( 30,197)( 31,199)
( 32,200)( 33,194)( 34,193)( 35,195)( 36,196)( 37,190)( 38,189)( 39,191)
( 40,192)( 41,226)( 42,225)( 43,227)( 44,228)( 45,222)( 46,221)( 47,223)
( 48,224)( 49,238)( 50,237)( 51,239)( 52,240)( 53,234)( 54,233)( 55,235)
( 56,236)( 57,230)( 58,229)( 59,231)( 60,232)( 61,326)( 62,325)( 63,327)
( 64,328)( 65,322)( 66,321)( 67,323)( 68,324)( 69,338)( 70,337)( 71,339)
( 72,340)( 73,334)( 74,333)( 75,335)( 76,336)( 77,330)( 78,329)( 79,331)
( 80,332)( 81,306)( 82,305)( 83,307)( 84,308)( 85,302)( 86,301)( 87,303)
( 88,304)( 89,318)( 90,317)( 91,319)( 92,320)( 93,314)( 94,313)( 95,315)
( 96,316)( 97,310)( 98,309)( 99,311)(100,312)(101,346)(102,345)(103,347)
(104,348)(105,342)(106,341)(107,343)(108,344)(109,358)(110,357)(111,359)
(112,360)(113,354)(114,353)(115,355)(116,356)(117,350)(118,349)(119,351)
(120,352)(121,266)(122,265)(123,267)(124,268)(125,262)(126,261)(127,263)
(128,264)(129,278)(130,277)(131,279)(132,280)(133,274)(134,273)(135,275)
(136,276)(137,270)(138,269)(139,271)(140,272)(141,246)(142,245)(143,247)
(144,248)(145,242)(146,241)(147,243)(148,244)(149,258)(150,257)(151,259)
(152,260)(153,254)(154,253)(155,255)(156,256)(157,250)(158,249)(159,251)
(160,252)(161,286)(162,285)(163,287)(164,288)(165,282)(166,281)(167,283)
(168,284)(169,298)(170,297)(171,299)(172,300)(173,294)(174,293)(175,295)
(176,296)(177,290)(178,289)(179,291)(180,292);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*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