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# Polytope of Type {6,24}

Atlas Canonical Name : {6,24}*864h
if this polytope has a name.
Group : SmallGroup(864,4303)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 18, 216, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,24,2} of size 1728
Vertex Figure Of :
{2,6,24} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*432i
3-fold quotients : {6,8}*288
4-fold quotients : {6,12}*216c
6-fold quotients : {6,4}*144
9-fold quotients : {2,24}*96
12-fold quotients : {6,4}*72
18-fold quotients : {2,12}*48
27-fold quotients : {2,8}*32
36-fold quotients : {2,6}*24
54-fold quotients : {2,4}*16
72-fold quotients : {2,3}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,48}*1728h, {12,24}*1728u
Permutation Representation (GAP) :
```s0 := (  1,163)(  2,164)(  3,165)(  4,169)(  5,170)(  6,171)(  7,166)(  8,167)
(  9,168)( 10,181)( 11,182)( 12,183)( 13,187)( 14,188)( 15,189)( 16,184)
( 17,185)( 18,186)( 19,172)( 20,173)( 21,174)( 22,178)( 23,179)( 24,180)
( 25,175)( 26,176)( 27,177)( 28,190)( 29,191)( 30,192)( 31,196)( 32,197)
( 33,198)( 34,193)( 35,194)( 36,195)( 37,208)( 38,209)( 39,210)( 40,214)
( 41,215)( 42,216)( 43,211)( 44,212)( 45,213)( 46,199)( 47,200)( 48,201)
( 49,205)( 50,206)( 51,207)( 52,202)( 53,203)( 54,204)( 55,136)( 56,137)
( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)( 64,154)
( 65,155)( 66,156)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)( 72,159)
( 73,145)( 74,146)( 75,147)( 76,151)( 77,152)( 78,153)( 79,148)( 80,149)
( 81,150)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)
( 89,113)( 90,114)( 91,127)( 92,128)( 93,129)( 94,133)( 95,134)( 96,135)
( 97,130)( 98,131)( 99,132)(100,118)(101,119)(102,120)(103,124)(104,125)
(105,126)(106,121)(107,122)(108,123)(217,379)(218,380)(219,381)(220,385)
(221,386)(222,387)(223,382)(224,383)(225,384)(226,397)(227,398)(228,399)
(229,403)(230,404)(231,405)(232,400)(233,401)(234,402)(235,388)(236,389)
(237,390)(238,394)(239,395)(240,396)(241,391)(242,392)(243,393)(244,406)
(245,407)(246,408)(247,412)(248,413)(249,414)(250,409)(251,410)(252,411)
(253,424)(254,425)(255,426)(256,430)(257,431)(258,432)(259,427)(260,428)
(261,429)(262,415)(263,416)(264,417)(265,421)(266,422)(267,423)(268,418)
(269,419)(270,420)(271,352)(272,353)(273,354)(274,358)(275,359)(276,360)
(277,355)(278,356)(279,357)(280,370)(281,371)(282,372)(283,376)(284,377)
(285,378)(286,373)(287,374)(288,375)(289,361)(290,362)(291,363)(292,367)
(293,368)(294,369)(295,364)(296,365)(297,366)(298,325)(299,326)(300,327)
(301,331)(302,332)(303,333)(304,328)(305,329)(306,330)(307,343)(308,344)
(309,345)(310,349)(311,350)(312,351)(313,346)(314,347)(315,348)(316,334)
(317,335)(318,336)(319,340)(320,341)(321,342)(322,337)(323,338)(324,339);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  5,  6)(  7, 25)(  8, 27)(  9, 26)( 13, 22)
( 14, 24)( 15, 23)( 17, 18)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 32, 33)
( 34, 52)( 35, 54)( 36, 53)( 40, 49)( 41, 51)( 42, 50)( 44, 45)( 47, 48)
( 55, 91)( 56, 93)( 57, 92)( 58, 85)( 59, 87)( 60, 86)( 61,106)( 62,108)
( 63,107)( 64, 82)( 65, 84)( 66, 83)( 67,103)( 68,105)( 69,104)( 70, 97)
( 71, 99)( 72, 98)( 73,100)( 74,102)( 75,101)( 76, 94)( 77, 96)( 78, 95)
( 79, 88)( 80, 90)( 81, 89)(109,145)(110,147)(111,146)(112,139)(113,141)
(114,140)(115,160)(116,162)(117,161)(118,136)(119,138)(120,137)(121,157)
(122,159)(123,158)(124,151)(125,153)(126,152)(127,154)(128,156)(129,155)
(130,148)(131,150)(132,149)(133,142)(134,144)(135,143)(163,172)(164,174)
(165,173)(167,168)(169,187)(170,189)(171,188)(175,184)(176,186)(177,185)
(179,180)(182,183)(190,199)(191,201)(192,200)(194,195)(196,214)(197,216)
(198,215)(202,211)(203,213)(204,212)(206,207)(209,210)(217,280)(218,282)
(219,281)(220,274)(221,276)(222,275)(223,295)(224,297)(225,296)(226,271)
(227,273)(228,272)(229,292)(230,294)(231,293)(232,286)(233,288)(234,287)
(235,289)(236,291)(237,290)(238,283)(239,285)(240,284)(241,277)(242,279)
(243,278)(244,307)(245,309)(246,308)(247,301)(248,303)(249,302)(250,322)
(251,324)(252,323)(253,298)(254,300)(255,299)(256,319)(257,321)(258,320)
(259,313)(260,315)(261,314)(262,316)(263,318)(264,317)(265,310)(266,312)
(267,311)(268,304)(269,306)(270,305)(325,415)(326,417)(327,416)(328,409)
(329,411)(330,410)(331,430)(332,432)(333,431)(334,406)(335,408)(336,407)
(337,427)(338,429)(339,428)(340,421)(341,423)(342,422)(343,424)(344,426)
(345,425)(346,418)(347,420)(348,419)(349,412)(350,414)(351,413)(352,388)
(353,390)(354,389)(355,382)(356,384)(357,383)(358,403)(359,405)(360,404)
(361,379)(362,381)(363,380)(364,400)(365,402)(366,401)(367,394)(368,396)
(369,395)(370,397)(371,399)(372,398)(373,391)(374,393)(375,392)(376,385)
(377,387)(378,386);;
s2 := (  1,218)(  2,217)(  3,219)(  4,224)(  5,223)(  6,225)(  7,221)(  8,220)
(  9,222)( 10,230)( 11,229)( 12,231)( 13,227)( 14,226)( 15,228)( 16,233)
( 17,232)( 18,234)( 19,242)( 20,241)( 21,243)( 22,239)( 23,238)( 24,240)
( 25,236)( 26,235)( 27,237)( 28,245)( 29,244)( 30,246)( 31,251)( 32,250)
( 33,252)( 34,248)( 35,247)( 36,249)( 37,257)( 38,256)( 39,258)( 40,254)
( 41,253)( 42,255)( 43,260)( 44,259)( 45,261)( 46,269)( 47,268)( 48,270)
( 49,266)( 50,265)( 51,267)( 52,263)( 53,262)( 54,264)( 55,299)( 56,298)
( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)( 64,311)
( 65,310)( 66,312)( 67,308)( 68,307)( 69,309)( 70,314)( 71,313)( 72,315)
( 73,323)( 74,322)( 75,324)( 76,320)( 77,319)( 78,321)( 79,317)( 80,316)
( 81,318)( 82,272)( 83,271)( 84,273)( 85,278)( 86,277)( 87,279)( 88,275)
( 89,274)( 90,276)( 91,284)( 92,283)( 93,285)( 94,281)( 95,280)( 96,282)
( 97,287)( 98,286)( 99,288)(100,296)(101,295)(102,297)(103,293)(104,292)
(105,294)(106,290)(107,289)(108,291)(109,353)(110,352)(111,354)(112,359)
(113,358)(114,360)(115,356)(116,355)(117,357)(118,365)(119,364)(120,366)
(121,362)(122,361)(123,363)(124,368)(125,367)(126,369)(127,377)(128,376)
(129,378)(130,374)(131,373)(132,375)(133,371)(134,370)(135,372)(136,326)
(137,325)(138,327)(139,332)(140,331)(141,333)(142,329)(143,328)(144,330)
(145,338)(146,337)(147,339)(148,335)(149,334)(150,336)(151,341)(152,340)
(153,342)(154,350)(155,349)(156,351)(157,347)(158,346)(159,348)(160,344)
(161,343)(162,345)(163,380)(164,379)(165,381)(166,386)(167,385)(168,387)
(169,383)(170,382)(171,384)(172,392)(173,391)(174,393)(175,389)(176,388)
(177,390)(178,395)(179,394)(180,396)(181,404)(182,403)(183,405)(184,401)
(185,400)(186,402)(187,398)(188,397)(189,399)(190,407)(191,406)(192,408)
(193,413)(194,412)(195,414)(196,410)(197,409)(198,411)(199,419)(200,418)
(201,420)(202,416)(203,415)(204,417)(205,422)(206,421)(207,423)(208,431)
(209,430)(210,432)(211,428)(212,427)(213,429)(214,425)(215,424)(216,426);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(432)!(  1,163)(  2,164)(  3,165)(  4,169)(  5,170)(  6,171)(  7,166)
(  8,167)(  9,168)( 10,181)( 11,182)( 12,183)( 13,187)( 14,188)( 15,189)
( 16,184)( 17,185)( 18,186)( 19,172)( 20,173)( 21,174)( 22,178)( 23,179)
( 24,180)( 25,175)( 26,176)( 27,177)( 28,190)( 29,191)( 30,192)( 31,196)
( 32,197)( 33,198)( 34,193)( 35,194)( 36,195)( 37,208)( 38,209)( 39,210)
( 40,214)( 41,215)( 42,216)( 43,211)( 44,212)( 45,213)( 46,199)( 47,200)
( 48,201)( 49,205)( 50,206)( 51,207)( 52,202)( 53,203)( 54,204)( 55,136)
( 56,137)( 57,138)( 58,142)( 59,143)( 60,144)( 61,139)( 62,140)( 63,141)
( 64,154)( 65,155)( 66,156)( 67,160)( 68,161)( 69,162)( 70,157)( 71,158)
( 72,159)( 73,145)( 74,146)( 75,147)( 76,151)( 77,152)( 78,153)( 79,148)
( 80,149)( 81,150)( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)
( 88,112)( 89,113)( 90,114)( 91,127)( 92,128)( 93,129)( 94,133)( 95,134)
( 96,135)( 97,130)( 98,131)( 99,132)(100,118)(101,119)(102,120)(103,124)
(104,125)(105,126)(106,121)(107,122)(108,123)(217,379)(218,380)(219,381)
(220,385)(221,386)(222,387)(223,382)(224,383)(225,384)(226,397)(227,398)
(228,399)(229,403)(230,404)(231,405)(232,400)(233,401)(234,402)(235,388)
(236,389)(237,390)(238,394)(239,395)(240,396)(241,391)(242,392)(243,393)
(244,406)(245,407)(246,408)(247,412)(248,413)(249,414)(250,409)(251,410)
(252,411)(253,424)(254,425)(255,426)(256,430)(257,431)(258,432)(259,427)
(260,428)(261,429)(262,415)(263,416)(264,417)(265,421)(266,422)(267,423)
(268,418)(269,419)(270,420)(271,352)(272,353)(273,354)(274,358)(275,359)
(276,360)(277,355)(278,356)(279,357)(280,370)(281,371)(282,372)(283,376)
(284,377)(285,378)(286,373)(287,374)(288,375)(289,361)(290,362)(291,363)
(292,367)(293,368)(294,369)(295,364)(296,365)(297,366)(298,325)(299,326)
(300,327)(301,331)(302,332)(303,333)(304,328)(305,329)(306,330)(307,343)
(308,344)(309,345)(310,349)(311,350)(312,351)(313,346)(314,347)(315,348)
(316,334)(317,335)(318,336)(319,340)(320,341)(321,342)(322,337)(323,338)
(324,339);
s1 := Sym(432)!(  1, 10)(  2, 12)(  3, 11)(  5,  6)(  7, 25)(  8, 27)(  9, 26)
( 13, 22)( 14, 24)( 15, 23)( 17, 18)( 20, 21)( 28, 37)( 29, 39)( 30, 38)
( 32, 33)( 34, 52)( 35, 54)( 36, 53)( 40, 49)( 41, 51)( 42, 50)( 44, 45)
( 47, 48)( 55, 91)( 56, 93)( 57, 92)( 58, 85)( 59, 87)( 60, 86)( 61,106)
( 62,108)( 63,107)( 64, 82)( 65, 84)( 66, 83)( 67,103)( 68,105)( 69,104)
( 70, 97)( 71, 99)( 72, 98)( 73,100)( 74,102)( 75,101)( 76, 94)( 77, 96)
( 78, 95)( 79, 88)( 80, 90)( 81, 89)(109,145)(110,147)(111,146)(112,139)
(113,141)(114,140)(115,160)(116,162)(117,161)(118,136)(119,138)(120,137)
(121,157)(122,159)(123,158)(124,151)(125,153)(126,152)(127,154)(128,156)
(129,155)(130,148)(131,150)(132,149)(133,142)(134,144)(135,143)(163,172)
(164,174)(165,173)(167,168)(169,187)(170,189)(171,188)(175,184)(176,186)
(177,185)(179,180)(182,183)(190,199)(191,201)(192,200)(194,195)(196,214)
(197,216)(198,215)(202,211)(203,213)(204,212)(206,207)(209,210)(217,280)
(218,282)(219,281)(220,274)(221,276)(222,275)(223,295)(224,297)(225,296)
(226,271)(227,273)(228,272)(229,292)(230,294)(231,293)(232,286)(233,288)
(234,287)(235,289)(236,291)(237,290)(238,283)(239,285)(240,284)(241,277)
(242,279)(243,278)(244,307)(245,309)(246,308)(247,301)(248,303)(249,302)
(250,322)(251,324)(252,323)(253,298)(254,300)(255,299)(256,319)(257,321)
(258,320)(259,313)(260,315)(261,314)(262,316)(263,318)(264,317)(265,310)
(266,312)(267,311)(268,304)(269,306)(270,305)(325,415)(326,417)(327,416)
(328,409)(329,411)(330,410)(331,430)(332,432)(333,431)(334,406)(335,408)
(336,407)(337,427)(338,429)(339,428)(340,421)(341,423)(342,422)(343,424)
(344,426)(345,425)(346,418)(347,420)(348,419)(349,412)(350,414)(351,413)
(352,388)(353,390)(354,389)(355,382)(356,384)(357,383)(358,403)(359,405)
(360,404)(361,379)(362,381)(363,380)(364,400)(365,402)(366,401)(367,394)
(368,396)(369,395)(370,397)(371,399)(372,398)(373,391)(374,393)(375,392)
(376,385)(377,387)(378,386);
s2 := Sym(432)!(  1,218)(  2,217)(  3,219)(  4,224)(  5,223)(  6,225)(  7,221)
(  8,220)(  9,222)( 10,230)( 11,229)( 12,231)( 13,227)( 14,226)( 15,228)
( 16,233)( 17,232)( 18,234)( 19,242)( 20,241)( 21,243)( 22,239)( 23,238)
( 24,240)( 25,236)( 26,235)( 27,237)( 28,245)( 29,244)( 30,246)( 31,251)
( 32,250)( 33,252)( 34,248)( 35,247)( 36,249)( 37,257)( 38,256)( 39,258)
( 40,254)( 41,253)( 42,255)( 43,260)( 44,259)( 45,261)( 46,269)( 47,268)
( 48,270)( 49,266)( 50,265)( 51,267)( 52,263)( 53,262)( 54,264)( 55,299)
( 56,298)( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)
( 64,311)( 65,310)( 66,312)( 67,308)( 68,307)( 69,309)( 70,314)( 71,313)
( 72,315)( 73,323)( 74,322)( 75,324)( 76,320)( 77,319)( 78,321)( 79,317)
( 80,316)( 81,318)( 82,272)( 83,271)( 84,273)( 85,278)( 86,277)( 87,279)
( 88,275)( 89,274)( 90,276)( 91,284)( 92,283)( 93,285)( 94,281)( 95,280)
( 96,282)( 97,287)( 98,286)( 99,288)(100,296)(101,295)(102,297)(103,293)
(104,292)(105,294)(106,290)(107,289)(108,291)(109,353)(110,352)(111,354)
(112,359)(113,358)(114,360)(115,356)(116,355)(117,357)(118,365)(119,364)
(120,366)(121,362)(122,361)(123,363)(124,368)(125,367)(126,369)(127,377)
(128,376)(129,378)(130,374)(131,373)(132,375)(133,371)(134,370)(135,372)
(136,326)(137,325)(138,327)(139,332)(140,331)(141,333)(142,329)(143,328)
(144,330)(145,338)(146,337)(147,339)(148,335)(149,334)(150,336)(151,341)
(152,340)(153,342)(154,350)(155,349)(156,351)(157,347)(158,346)(159,348)
(160,344)(161,343)(162,345)(163,380)(164,379)(165,381)(166,386)(167,385)
(168,387)(169,383)(170,382)(171,384)(172,392)(173,391)(174,393)(175,389)
(176,388)(177,390)(178,395)(179,394)(180,396)(181,404)(182,403)(183,405)
(184,401)(185,400)(186,402)(187,398)(188,397)(189,399)(190,407)(191,406)
(192,408)(193,413)(194,412)(195,414)(196,410)(197,409)(198,411)(199,419)
(200,418)(201,420)(202,416)(203,415)(204,417)(205,422)(206,421)(207,423)
(208,431)(209,430)(210,432)(211,428)(212,427)(213,429)(214,425)(215,424)
(216,426);
poly := sub<Sym(432)|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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*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 >;

```
References : None.
to this polytope