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# Polytope of Type {24,6}

Atlas Canonical Name : {24,6}*864d
if this polytope has a name.
Group : SmallGroup(864,2265)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 72, 216, 18
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{24,6,2} of size 1728
Vertex Figure Of :
{2,24,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6}*432e
3-fold quotients : {8,6}*288
4-fold quotients : {12,6}*216b
6-fold quotients : {4,6}*144
12-fold quotients : {4,6}*72
27-fold quotients : {8,2}*32
54-fold quotients : {4,2}*16
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {48,6}*1728e, {24,12}*1728h
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  5)(  7,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)
( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 32)( 34, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 50)( 41, 49)( 42, 51)( 43, 54)( 44, 53)( 45, 52)
( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)
( 63, 88)( 64,100)( 65,102)( 66,101)( 67,104)( 68,103)( 69,105)( 70,108)
( 71,107)( 72,106)( 73, 91)( 74, 93)( 75, 92)( 76, 95)( 77, 94)( 78, 96)
( 79, 99)( 80, 98)( 81, 97)(109,136)(110,138)(111,137)(112,140)(113,139)
(114,141)(115,144)(116,143)(117,142)(118,154)(119,156)(120,155)(121,158)
(122,157)(123,159)(124,162)(125,161)(126,160)(127,145)(128,147)(129,146)
(130,149)(131,148)(132,150)(133,153)(134,152)(135,151)(164,165)(166,167)
(169,171)(172,181)(173,183)(174,182)(175,185)(176,184)(177,186)(178,189)
(179,188)(180,187)(191,192)(193,194)(196,198)(199,208)(200,210)(201,209)
(202,212)(203,211)(204,213)(205,216)(206,215)(207,214)(217,271)(218,273)
(219,272)(220,275)(221,274)(222,276)(223,279)(224,278)(225,277)(226,289)
(227,291)(228,290)(229,293)(230,292)(231,294)(232,297)(233,296)(234,295)
(235,280)(236,282)(237,281)(238,284)(239,283)(240,285)(241,288)(242,287)
(243,286)(244,298)(245,300)(246,299)(247,302)(248,301)(249,303)(250,306)
(251,305)(252,304)(253,316)(254,318)(255,317)(256,320)(257,319)(258,321)
(259,324)(260,323)(261,322)(262,307)(263,309)(264,308)(265,311)(266,310)
(267,312)(268,315)(269,314)(270,313)(325,406)(326,408)(327,407)(328,410)
(329,409)(330,411)(331,414)(332,413)(333,412)(334,424)(335,426)(336,425)
(337,428)(338,427)(339,429)(340,432)(341,431)(342,430)(343,415)(344,417)
(345,416)(346,419)(347,418)(348,420)(349,423)(350,422)(351,421)(352,379)
(353,381)(354,380)(355,383)(356,382)(357,384)(358,387)(359,386)(360,385)
(361,397)(362,399)(363,398)(364,401)(365,400)(366,402)(367,405)(368,404)
(369,403)(370,388)(371,390)(372,389)(373,392)(374,391)(375,393)(376,396)
(377,395)(378,394);;
s1 := (  1,218)(  2,217)(  3,219)(  4,226)(  5,228)(  6,227)(  7,237)(  8,236)
(  9,235)( 10,220)( 11,222)( 12,221)( 13,229)( 14,231)( 15,230)( 16,238)
( 17,240)( 18,239)( 19,225)( 20,224)( 21,223)( 22,232)( 23,234)( 24,233)
( 25,242)( 26,241)( 27,243)( 28,245)( 29,244)( 30,246)( 31,253)( 32,255)
( 33,254)( 34,264)( 35,263)( 36,262)( 37,247)( 38,249)( 39,248)( 40,256)
( 41,258)( 42,257)( 43,265)( 44,267)( 45,266)( 46,252)( 47,251)( 48,250)
( 49,259)( 50,261)( 51,260)( 52,269)( 53,268)( 54,270)( 55,299)( 56,298)
( 57,300)( 58,307)( 59,309)( 60,308)( 61,318)( 62,317)( 63,316)( 64,301)
( 65,303)( 66,302)( 67,310)( 68,312)( 69,311)( 70,319)( 71,321)( 72,320)
( 73,306)( 74,305)( 75,304)( 76,313)( 77,315)( 78,314)( 79,323)( 80,322)
( 81,324)( 82,272)( 83,271)( 84,273)( 85,280)( 86,282)( 87,281)( 88,291)
( 89,290)( 90,289)( 91,274)( 92,276)( 93,275)( 94,283)( 95,285)( 96,284)
( 97,292)( 98,294)( 99,293)(100,279)(101,278)(102,277)(103,286)(104,288)
(105,287)(106,296)(107,295)(108,297)(109,353)(110,352)(111,354)(112,361)
(113,363)(114,362)(115,372)(116,371)(117,370)(118,355)(119,357)(120,356)
(121,364)(122,366)(123,365)(124,373)(125,375)(126,374)(127,360)(128,359)
(129,358)(130,367)(131,369)(132,368)(133,377)(134,376)(135,378)(136,326)
(137,325)(138,327)(139,334)(140,336)(141,335)(142,345)(143,344)(144,343)
(145,328)(146,330)(147,329)(148,337)(149,339)(150,338)(151,346)(152,348)
(153,347)(154,333)(155,332)(156,331)(157,340)(158,342)(159,341)(160,350)
(161,349)(162,351)(163,380)(164,379)(165,381)(166,388)(167,390)(168,389)
(169,399)(170,398)(171,397)(172,382)(173,384)(174,383)(175,391)(176,393)
(177,392)(178,400)(179,402)(180,401)(181,387)(182,386)(183,385)(184,394)
(185,396)(186,395)(187,404)(188,403)(189,405)(190,407)(191,406)(192,408)
(193,415)(194,417)(195,416)(196,426)(197,425)(198,424)(199,409)(200,411)
(201,410)(202,418)(203,420)(204,419)(205,427)(206,429)(207,428)(208,414)
(209,413)(210,412)(211,421)(212,423)(213,422)(214,431)(215,430)(216,432);;
s2 := (  1,170)(  2,171)(  3,169)(  4,166)(  5,167)(  6,168)(  7,165)(  8,163)
(  9,164)( 10,188)( 11,189)( 12,187)( 13,184)( 14,185)( 15,186)( 16,183)
( 17,181)( 18,182)( 19,179)( 20,180)( 21,178)( 22,175)( 23,176)( 24,177)
( 25,174)( 26,172)( 27,173)( 28,197)( 29,198)( 30,196)( 31,193)( 32,194)
( 33,195)( 34,192)( 35,190)( 36,191)( 37,215)( 38,216)( 39,214)( 40,211)
( 41,212)( 42,213)( 43,210)( 44,208)( 45,209)( 46,206)( 47,207)( 48,205)
( 49,202)( 50,203)( 51,204)( 52,201)( 53,199)( 54,200)( 55,143)( 56,144)
( 57,142)( 58,139)( 59,140)( 60,141)( 61,138)( 62,136)( 63,137)( 64,161)
( 65,162)( 66,160)( 67,157)( 68,158)( 69,159)( 70,156)( 71,154)( 72,155)
( 73,152)( 74,153)( 75,151)( 76,148)( 77,149)( 78,150)( 79,147)( 80,145)
( 81,146)( 82,116)( 83,117)( 84,115)( 85,112)( 86,113)( 87,114)( 88,111)
( 89,109)( 90,110)( 91,134)( 92,135)( 93,133)( 94,130)( 95,131)( 96,132)
( 97,129)( 98,127)( 99,128)(100,125)(101,126)(102,124)(103,121)(104,122)
(105,123)(106,120)(107,118)(108,119)(217,386)(218,387)(219,385)(220,382)
(221,383)(222,384)(223,381)(224,379)(225,380)(226,404)(227,405)(228,403)
(229,400)(230,401)(231,402)(232,399)(233,397)(234,398)(235,395)(236,396)
(237,394)(238,391)(239,392)(240,393)(241,390)(242,388)(243,389)(244,413)
(245,414)(246,412)(247,409)(248,410)(249,411)(250,408)(251,406)(252,407)
(253,431)(254,432)(255,430)(256,427)(257,428)(258,429)(259,426)(260,424)
(261,425)(262,422)(263,423)(264,421)(265,418)(266,419)(267,420)(268,417)
(269,415)(270,416)(271,359)(272,360)(273,358)(274,355)(275,356)(276,357)
(277,354)(278,352)(279,353)(280,377)(281,378)(282,376)(283,373)(284,374)
(285,375)(286,372)(287,370)(288,371)(289,368)(290,369)(291,367)(292,364)
(293,365)(294,366)(295,363)(296,361)(297,362)(298,332)(299,333)(300,331)
(301,328)(302,329)(303,330)(304,327)(305,325)(306,326)(307,350)(308,351)
(309,349)(310,346)(311,347)(312,348)(313,345)(314,343)(315,344)(316,341)
(317,342)(318,340)(319,337)(320,338)(321,339)(322,336)(323,334)(324,335);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope