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# Polytope of Type {24,6}

Atlas Canonical Name : {24,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30272)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 144, 432, 36
Order of s0s1s2 : 6
Order of s0s1s2s1 : 24
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,6}*864a
3-fold quotients : {24,6}*576d
4-fold quotients : {12,3}*432
6-fold quotients : {12,6}*288b
8-fold quotients : {6,6}*216a
9-fold quotients : {8,6}*192c
12-fold quotients : {12,3}*144
16-fold quotients : {6,3}*108
18-fold quotients : {4,6}*96
24-fold quotients : {6,6}*72b
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {6,3}*36
72-fold quotients : {4,3}*24, {2,6}*24
144-fold quotients : {2,3}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)
(  9,237)( 10,238)( 11,240)( 12,239)( 13,234)( 14,233)( 15,235)( 16,236)
( 17,229)( 18,230)( 19,232)( 20,231)( 21,226)( 22,225)( 23,227)( 24,228)
( 25,269)( 26,270)( 27,272)( 28,271)( 29,266)( 30,265)( 31,267)( 32,268)
( 33,285)( 34,286)( 35,288)( 36,287)( 37,282)( 38,281)( 39,283)( 40,284)
( 41,277)( 42,278)( 43,280)( 44,279)( 45,274)( 46,273)( 47,275)( 48,276)
( 49,245)( 50,246)( 51,248)( 52,247)( 53,242)( 54,241)( 55,243)( 56,244)
( 57,261)( 58,262)( 59,264)( 60,263)( 61,258)( 62,257)( 63,259)( 64,260)
( 65,253)( 66,254)( 67,256)( 68,255)( 69,250)( 70,249)( 71,251)( 72,252)
( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)
( 81,309)( 82,310)( 83,312)( 84,311)( 85,306)( 86,305)( 87,307)( 88,308)
( 89,301)( 90,302)( 91,304)( 92,303)( 93,298)( 94,297)( 95,299)( 96,300)
( 97,341)( 98,342)( 99,344)(100,343)(101,338)(102,337)(103,339)(104,340)
(105,357)(106,358)(107,360)(108,359)(109,354)(110,353)(111,355)(112,356)
(113,349)(114,350)(115,352)(116,351)(117,346)(118,345)(119,347)(120,348)
(121,317)(122,318)(123,320)(124,319)(125,314)(126,313)(127,315)(128,316)
(129,333)(130,334)(131,336)(132,335)(133,330)(134,329)(135,331)(136,332)
(137,325)(138,326)(139,328)(140,327)(141,322)(142,321)(143,323)(144,324)
(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)
(153,381)(154,382)(155,384)(156,383)(157,378)(158,377)(159,379)(160,380)
(161,373)(162,374)(163,376)(164,375)(165,370)(166,369)(167,371)(168,372)
(169,413)(170,414)(171,416)(172,415)(173,410)(174,409)(175,411)(176,412)
(177,429)(178,430)(179,432)(180,431)(181,426)(182,425)(183,427)(184,428)
(185,421)(186,422)(187,424)(188,423)(189,418)(190,417)(191,419)(192,420)
(193,389)(194,390)(195,392)(196,391)(197,386)(198,385)(199,387)(200,388)
(201,405)(202,406)(203,408)(204,407)(205,402)(206,401)(207,403)(208,404)
(209,397)(210,398)(211,400)(212,399)(213,394)(214,393)(215,395)(216,396);;
s1 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 31)(  6, 32)(  7, 29)(  8, 30)
(  9, 33)( 10, 34)( 11, 36)( 12, 35)( 13, 39)( 14, 40)( 15, 37)( 16, 38)
( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 47)( 22, 48)( 23, 45)( 24, 46)
( 51, 52)( 53, 55)( 54, 56)( 59, 60)( 61, 63)( 62, 64)( 67, 68)( 69, 71)
( 70, 72)( 73,169)( 74,170)( 75,172)( 76,171)( 77,175)( 78,176)( 79,173)
( 80,174)( 81,177)( 82,178)( 83,180)( 84,179)( 85,183)( 86,184)( 87,181)
( 88,182)( 89,185)( 90,186)( 91,188)( 92,187)( 93,191)( 94,192)( 95,189)
( 96,190)( 97,145)( 98,146)( 99,148)(100,147)(101,151)(102,152)(103,149)
(104,150)(105,153)(106,154)(107,156)(108,155)(109,159)(110,160)(111,157)
(112,158)(113,161)(114,162)(115,164)(116,163)(117,167)(118,168)(119,165)
(120,166)(121,193)(122,194)(123,196)(124,195)(125,199)(126,200)(127,197)
(128,198)(129,201)(130,202)(131,204)(132,203)(133,207)(134,208)(135,205)
(136,206)(137,209)(138,210)(139,212)(140,211)(141,215)(142,216)(143,213)
(144,214)(217,241)(218,242)(219,244)(220,243)(221,247)(222,248)(223,245)
(224,246)(225,249)(226,250)(227,252)(228,251)(229,255)(230,256)(231,253)
(232,254)(233,257)(234,258)(235,260)(236,259)(237,263)(238,264)(239,261)
(240,262)(267,268)(269,271)(270,272)(275,276)(277,279)(278,280)(283,284)
(285,287)(286,288)(289,385)(290,386)(291,388)(292,387)(293,391)(294,392)
(295,389)(296,390)(297,393)(298,394)(299,396)(300,395)(301,399)(302,400)
(303,397)(304,398)(305,401)(306,402)(307,404)(308,403)(309,407)(310,408)
(311,405)(312,406)(313,361)(314,362)(315,364)(316,363)(317,367)(318,368)
(319,365)(320,366)(321,369)(322,370)(323,372)(324,371)(325,375)(326,376)
(327,373)(328,374)(329,377)(330,378)(331,380)(332,379)(333,383)(334,384)
(335,381)(336,382)(337,409)(338,410)(339,412)(340,411)(341,415)(342,416)
(343,413)(344,414)(345,417)(346,418)(347,420)(348,419)(349,423)(350,424)
(351,421)(352,422)(353,425)(354,426)(355,428)(356,427)(357,431)(358,432)
(359,429)(360,430);;
s2 := (  1,289)(  2,290)(  3,295)(  4,296)(  5,294)(  6,293)(  7,291)(  8,292)
(  9,297)( 10,298)( 11,303)( 12,304)( 13,302)( 14,301)( 15,299)( 16,300)
( 17,305)( 18,306)( 19,311)( 20,312)( 21,310)( 22,309)( 23,307)( 24,308)
( 25,353)( 26,354)( 27,359)( 28,360)( 29,358)( 30,357)( 31,355)( 32,356)
( 33,337)( 34,338)( 35,343)( 36,344)( 37,342)( 38,341)( 39,339)( 40,340)
( 41,345)( 42,346)( 43,351)( 44,352)( 45,350)( 46,349)( 47,347)( 48,348)
( 49,321)( 50,322)( 51,327)( 52,328)( 53,326)( 54,325)( 55,323)( 56,324)
( 57,329)( 58,330)( 59,335)( 60,336)( 61,334)( 62,333)( 63,331)( 64,332)
( 65,313)( 66,314)( 67,319)( 68,320)( 69,318)( 70,317)( 71,315)( 72,316)
( 73,217)( 74,218)( 75,223)( 76,224)( 77,222)( 78,221)( 79,219)( 80,220)
( 81,225)( 82,226)( 83,231)( 84,232)( 85,230)( 86,229)( 87,227)( 88,228)
( 89,233)( 90,234)( 91,239)( 92,240)( 93,238)( 94,237)( 95,235)( 96,236)
( 97,281)( 98,282)( 99,287)(100,288)(101,286)(102,285)(103,283)(104,284)
(105,265)(106,266)(107,271)(108,272)(109,270)(110,269)(111,267)(112,268)
(113,273)(114,274)(115,279)(116,280)(117,278)(118,277)(119,275)(120,276)
(121,249)(122,250)(123,255)(124,256)(125,254)(126,253)(127,251)(128,252)
(129,257)(130,258)(131,263)(132,264)(133,262)(134,261)(135,259)(136,260)
(137,241)(138,242)(139,247)(140,248)(141,246)(142,245)(143,243)(144,244)
(145,361)(146,362)(147,367)(148,368)(149,366)(150,365)(151,363)(152,364)
(153,369)(154,370)(155,375)(156,376)(157,374)(158,373)(159,371)(160,372)
(161,377)(162,378)(163,383)(164,384)(165,382)(166,381)(167,379)(168,380)
(169,425)(170,426)(171,431)(172,432)(173,430)(174,429)(175,427)(176,428)
(177,409)(178,410)(179,415)(180,416)(181,414)(182,413)(183,411)(184,412)
(185,417)(186,418)(187,423)(188,424)(189,422)(190,421)(191,419)(192,420)
(193,393)(194,394)(195,399)(196,400)(197,398)(198,397)(199,395)(200,396)
(201,401)(202,402)(203,407)(204,408)(205,406)(206,405)(207,403)(208,404)
(209,385)(210,386)(211,391)(212,392)(213,390)(214,389)(215,387)(216,388);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope