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# Polytope of Type {24,18}

Atlas Canonical Name : {24,18}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30258)
Rank : 3
Schlafli Type : {24,18}
Number of vertices, edges, etc : 48, 432, 36
Order of s0s1s2 : 18
Order of s0s1s2s1 : 24
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 : {12,18}*864b
3-fold quotients : {8,18}*576c, {24,6}*576d
4-fold quotients : {12,9}*432
6-fold quotients : {4,18}*288, {12,6}*288b
8-fold quotients : {6,18}*216b
9-fold quotients : {8,6}*192c
12-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c, {12,3}*144
16-fold quotients : {6,9}*108
18-fold quotients : {4,6}*96
24-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72b
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,9}*36, {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)
(  9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)( 16,228)
( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)( 24,236)
( 25,269)( 26,270)( 27,272)( 28,271)( 29,266)( 30,265)( 31,267)( 32,268)
( 33,277)( 34,278)( 35,280)( 36,279)( 37,274)( 38,273)( 39,275)( 40,276)
( 41,285)( 42,286)( 43,288)( 44,287)( 45,282)( 46,281)( 47,283)( 48,284)
( 49,245)( 50,246)( 51,248)( 52,247)( 53,242)( 54,241)( 55,243)( 56,244)
( 57,253)( 58,254)( 59,256)( 60,255)( 61,250)( 62,249)( 63,251)( 64,252)
( 65,261)( 66,262)( 67,264)( 68,263)( 69,258)( 70,257)( 71,259)( 72,260)
( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)
( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)( 88,300)
( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)( 96,308)
( 97,341)( 98,342)( 99,344)(100,343)(101,338)(102,337)(103,339)(104,340)
(105,349)(106,350)(107,352)(108,351)(109,346)(110,345)(111,347)(112,348)
(113,357)(114,358)(115,360)(116,359)(117,354)(118,353)(119,355)(120,356)
(121,317)(122,318)(123,320)(124,319)(125,314)(126,313)(127,315)(128,316)
(129,325)(130,326)(131,328)(132,327)(133,322)(134,321)(135,323)(136,324)
(137,333)(138,334)(139,336)(140,335)(141,330)(142,329)(143,331)(144,332)
(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)
(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)(160,372)
(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)
(169,413)(170,414)(171,416)(172,415)(173,410)(174,409)(175,411)(176,412)
(177,421)(178,422)(179,424)(180,423)(181,418)(182,417)(183,419)(184,420)
(185,429)(186,430)(187,432)(188,431)(189,426)(190,425)(191,427)(192,428)
(193,389)(194,390)(195,392)(196,391)(197,386)(198,385)(199,387)(200,388)
(201,397)(202,398)(203,400)(204,399)(205,394)(206,393)(207,395)(208,396)
(209,405)(210,406)(211,408)(212,407)(213,402)(214,401)(215,403)(216,404);;
s1 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 31)(  6, 32)(  7, 29)(  8, 30)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 39)( 22, 40)( 23, 37)( 24, 38)
( 51, 52)( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)
( 62, 72)( 63, 69)( 64, 70)( 73,185)( 74,186)( 75,188)( 76,187)( 77,191)
( 78,192)( 79,189)( 80,190)( 81,177)( 82,178)( 83,180)( 84,179)( 85,183)
( 86,184)( 87,181)( 88,182)( 89,169)( 90,170)( 91,172)( 92,171)( 93,175)
( 94,176)( 95,173)( 96,174)( 97,161)( 98,162)( 99,164)(100,163)(101,167)
(102,168)(103,165)(104,166)(105,153)(106,154)(107,156)(108,155)(109,159)
(110,160)(111,157)(112,158)(113,145)(114,146)(115,148)(116,147)(117,151)
(118,152)(119,149)(120,150)(121,209)(122,210)(123,212)(124,211)(125,215)
(126,216)(127,213)(128,214)(129,201)(130,202)(131,204)(132,203)(133,207)
(134,208)(135,205)(136,206)(137,193)(138,194)(139,196)(140,195)(141,199)
(142,200)(143,197)(144,198)(217,241)(218,242)(219,244)(220,243)(221,247)
(222,248)(223,245)(224,246)(225,257)(226,258)(227,260)(228,259)(229,263)
(230,264)(231,261)(232,262)(233,249)(234,250)(235,252)(236,251)(237,255)
(238,256)(239,253)(240,254)(267,268)(269,271)(270,272)(273,281)(274,282)
(275,284)(276,283)(277,287)(278,288)(279,285)(280,286)(289,401)(290,402)
(291,404)(292,403)(293,407)(294,408)(295,405)(296,406)(297,393)(298,394)
(299,396)(300,395)(301,399)(302,400)(303,397)(304,398)(305,385)(306,386)
(307,388)(308,387)(309,391)(310,392)(311,389)(312,390)(313,377)(314,378)
(315,380)(316,379)(317,383)(318,384)(319,381)(320,382)(321,369)(322,370)
(323,372)(324,371)(325,375)(326,376)(327,373)(328,374)(329,361)(330,362)
(331,364)(332,363)(333,367)(334,368)(335,365)(336,366)(337,425)(338,426)
(339,428)(340,427)(341,431)(342,432)(343,429)(344,430)(345,417)(346,418)
(347,420)(348,419)(349,423)(350,424)(351,421)(352,422)(353,409)(354,410)
(355,412)(356,411)(357,415)(358,416)(359,413)(360,414);;
s2 := (  1,289)(  2,290)(  3,295)(  4,296)(  5,294)(  6,293)(  7,291)(  8,292)
(  9,305)( 10,306)( 11,311)( 12,312)( 13,310)( 14,309)( 15,307)( 16,308)
( 17,297)( 18,298)( 19,303)( 20,304)( 21,302)( 22,301)( 23,299)( 24,300)
( 25,337)( 26,338)( 27,343)( 28,344)( 29,342)( 30,341)( 31,339)( 32,340)
( 33,353)( 34,354)( 35,359)( 36,360)( 37,358)( 38,357)( 39,355)( 40,356)
( 41,345)( 42,346)( 43,351)( 44,352)( 45,350)( 46,349)( 47,347)( 48,348)
( 49,313)( 50,314)( 51,319)( 52,320)( 53,318)( 54,317)( 55,315)( 56,316)
( 57,329)( 58,330)( 59,335)( 60,336)( 61,334)( 62,333)( 63,331)( 64,332)
( 65,321)( 66,322)( 67,327)( 68,328)( 69,326)( 70,325)( 71,323)( 72,324)
( 73,217)( 74,218)( 75,223)( 76,224)( 77,222)( 78,221)( 79,219)( 80,220)
( 81,233)( 82,234)( 83,239)( 84,240)( 85,238)( 86,237)( 87,235)( 88,236)
( 89,225)( 90,226)( 91,231)( 92,232)( 93,230)( 94,229)( 95,227)( 96,228)
( 97,265)( 98,266)( 99,271)(100,272)(101,270)(102,269)(103,267)(104,268)
(105,281)(106,282)(107,287)(108,288)(109,286)(110,285)(111,283)(112,284)
(113,273)(114,274)(115,279)(116,280)(117,278)(118,277)(119,275)(120,276)
(121,241)(122,242)(123,247)(124,248)(125,246)(126,245)(127,243)(128,244)
(129,257)(130,258)(131,263)(132,264)(133,262)(134,261)(135,259)(136,260)
(137,249)(138,250)(139,255)(140,256)(141,254)(142,253)(143,251)(144,252)
(145,377)(146,378)(147,383)(148,384)(149,382)(150,381)(151,379)(152,380)
(153,369)(154,370)(155,375)(156,376)(157,374)(158,373)(159,371)(160,372)
(161,361)(162,362)(163,367)(164,368)(165,366)(166,365)(167,363)(168,364)
(169,425)(170,426)(171,431)(172,432)(173,430)(174,429)(175,427)(176,428)
(177,417)(178,418)(179,423)(180,424)(181,422)(182,421)(183,419)(184,420)
(185,409)(186,410)(187,415)(188,416)(189,414)(190,413)(191,411)(192,412)
(193,401)(194,402)(195,407)(196,408)(197,406)(198,405)(199,403)(200,404)
(201,393)(202,394)(203,399)(204,400)(205,398)(206,397)(207,395)(208,396)
(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, s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*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 ];;
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)(  9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)
( 16,228)( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)
( 24,236)( 25,269)( 26,270)( 27,272)( 28,271)( 29,266)( 30,265)( 31,267)
( 32,268)( 33,277)( 34,278)( 35,280)( 36,279)( 37,274)( 38,273)( 39,275)
( 40,276)( 41,285)( 42,286)( 43,288)( 44,287)( 45,282)( 46,281)( 47,283)
( 48,284)( 49,245)( 50,246)( 51,248)( 52,247)( 53,242)( 54,241)( 55,243)
( 56,244)( 57,253)( 58,254)( 59,256)( 60,255)( 61,250)( 62,249)( 63,251)
( 64,252)( 65,261)( 66,262)( 67,264)( 68,263)( 69,258)( 70,257)( 71,259)
( 72,260)( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)
( 80,292)( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)
( 88,300)( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)
( 96,308)( 97,341)( 98,342)( 99,344)(100,343)(101,338)(102,337)(103,339)
(104,340)(105,349)(106,350)(107,352)(108,351)(109,346)(110,345)(111,347)
(112,348)(113,357)(114,358)(115,360)(116,359)(117,354)(118,353)(119,355)
(120,356)(121,317)(122,318)(123,320)(124,319)(125,314)(126,313)(127,315)
(128,316)(129,325)(130,326)(131,328)(132,327)(133,322)(134,321)(135,323)
(136,324)(137,333)(138,334)(139,336)(140,335)(141,330)(142,329)(143,331)
(144,332)(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)
(152,364)(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)
(160,372)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)
(168,380)(169,413)(170,414)(171,416)(172,415)(173,410)(174,409)(175,411)
(176,412)(177,421)(178,422)(179,424)(180,423)(181,418)(182,417)(183,419)
(184,420)(185,429)(186,430)(187,432)(188,431)(189,426)(190,425)(191,427)
(192,428)(193,389)(194,390)(195,392)(196,391)(197,386)(198,385)(199,387)
(200,388)(201,397)(202,398)(203,400)(204,399)(205,394)(206,393)(207,395)
(208,396)(209,405)(210,406)(211,408)(212,407)(213,402)(214,401)(215,403)
(216,404);
s1 := Sym(432)!(  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 31)(  6, 32)(  7, 29)
(  8, 30)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)
( 16, 46)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 39)( 22, 40)( 23, 37)
( 24, 38)( 51, 52)( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)
( 61, 71)( 62, 72)( 63, 69)( 64, 70)( 73,185)( 74,186)( 75,188)( 76,187)
( 77,191)( 78,192)( 79,189)( 80,190)( 81,177)( 82,178)( 83,180)( 84,179)
( 85,183)( 86,184)( 87,181)( 88,182)( 89,169)( 90,170)( 91,172)( 92,171)
( 93,175)( 94,176)( 95,173)( 96,174)( 97,161)( 98,162)( 99,164)(100,163)
(101,167)(102,168)(103,165)(104,166)(105,153)(106,154)(107,156)(108,155)
(109,159)(110,160)(111,157)(112,158)(113,145)(114,146)(115,148)(116,147)
(117,151)(118,152)(119,149)(120,150)(121,209)(122,210)(123,212)(124,211)
(125,215)(126,216)(127,213)(128,214)(129,201)(130,202)(131,204)(132,203)
(133,207)(134,208)(135,205)(136,206)(137,193)(138,194)(139,196)(140,195)
(141,199)(142,200)(143,197)(144,198)(217,241)(218,242)(219,244)(220,243)
(221,247)(222,248)(223,245)(224,246)(225,257)(226,258)(227,260)(228,259)
(229,263)(230,264)(231,261)(232,262)(233,249)(234,250)(235,252)(236,251)
(237,255)(238,256)(239,253)(240,254)(267,268)(269,271)(270,272)(273,281)
(274,282)(275,284)(276,283)(277,287)(278,288)(279,285)(280,286)(289,401)
(290,402)(291,404)(292,403)(293,407)(294,408)(295,405)(296,406)(297,393)
(298,394)(299,396)(300,395)(301,399)(302,400)(303,397)(304,398)(305,385)
(306,386)(307,388)(308,387)(309,391)(310,392)(311,389)(312,390)(313,377)
(314,378)(315,380)(316,379)(317,383)(318,384)(319,381)(320,382)(321,369)
(322,370)(323,372)(324,371)(325,375)(326,376)(327,373)(328,374)(329,361)
(330,362)(331,364)(332,363)(333,367)(334,368)(335,365)(336,366)(337,425)
(338,426)(339,428)(340,427)(341,431)(342,432)(343,429)(344,430)(345,417)
(346,418)(347,420)(348,419)(349,423)(350,424)(351,421)(352,422)(353,409)
(354,410)(355,412)(356,411)(357,415)(358,416)(359,413)(360,414);
s2 := Sym(432)!(  1,289)(  2,290)(  3,295)(  4,296)(  5,294)(  6,293)(  7,291)
(  8,292)(  9,305)( 10,306)( 11,311)( 12,312)( 13,310)( 14,309)( 15,307)
( 16,308)( 17,297)( 18,298)( 19,303)( 20,304)( 21,302)( 22,301)( 23,299)
( 24,300)( 25,337)( 26,338)( 27,343)( 28,344)( 29,342)( 30,341)( 31,339)
( 32,340)( 33,353)( 34,354)( 35,359)( 36,360)( 37,358)( 38,357)( 39,355)
( 40,356)( 41,345)( 42,346)( 43,351)( 44,352)( 45,350)( 46,349)( 47,347)
( 48,348)( 49,313)( 50,314)( 51,319)( 52,320)( 53,318)( 54,317)( 55,315)
( 56,316)( 57,329)( 58,330)( 59,335)( 60,336)( 61,334)( 62,333)( 63,331)
( 64,332)( 65,321)( 66,322)( 67,327)( 68,328)( 69,326)( 70,325)( 71,323)
( 72,324)( 73,217)( 74,218)( 75,223)( 76,224)( 77,222)( 78,221)( 79,219)
( 80,220)( 81,233)( 82,234)( 83,239)( 84,240)( 85,238)( 86,237)( 87,235)
( 88,236)( 89,225)( 90,226)( 91,231)( 92,232)( 93,230)( 94,229)( 95,227)
( 96,228)( 97,265)( 98,266)( 99,271)(100,272)(101,270)(102,269)(103,267)
(104,268)(105,281)(106,282)(107,287)(108,288)(109,286)(110,285)(111,283)
(112,284)(113,273)(114,274)(115,279)(116,280)(117,278)(118,277)(119,275)
(120,276)(121,241)(122,242)(123,247)(124,248)(125,246)(126,245)(127,243)
(128,244)(129,257)(130,258)(131,263)(132,264)(133,262)(134,261)(135,259)
(136,260)(137,249)(138,250)(139,255)(140,256)(141,254)(142,253)(143,251)
(144,252)(145,377)(146,378)(147,383)(148,384)(149,382)(150,381)(151,379)
(152,380)(153,369)(154,370)(155,375)(156,376)(157,374)(158,373)(159,371)
(160,372)(161,361)(162,362)(163,367)(164,368)(165,366)(166,365)(167,363)
(168,364)(169,425)(170,426)(171,431)(172,432)(173,430)(174,429)(175,427)
(176,428)(177,417)(178,418)(179,423)(180,424)(181,422)(182,421)(183,419)
(184,420)(185,409)(186,410)(187,415)(188,416)(189,414)(190,413)(191,411)
(192,412)(193,401)(194,402)(195,407)(196,408)(197,406)(198,405)(199,403)
(200,404)(201,393)(202,394)(203,399)(204,400)(205,398)(206,397)(207,395)
(208,396)(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, s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*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 >;

```
References : None.
to this polytope