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# Polytope of Type {18,24}

Atlas Canonical Name : {18,24}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30284)
Rank : 3
Schlafli Type : {18,24}
Number of vertices, edges, etc : 36, 432, 48
Order of s0s1s2 : 36
Order of s0s1s2s1 : 8
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 : {18,12}*864a
3-fold quotients : {18,8}*576b, {6,24}*576e
4-fold quotients : {18,12}*432c
6-fold quotients : {9,8}*288, {18,4}*288, {6,12}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {6,8}*192b
12-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c, {6,12}*144d
18-fold quotients : {3,8}*96, {6,4}*96
24-fold quotients : {9,4}*72, {18,2}*72, {6,6}*72a
36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {9,2}*36
72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
144-fold quotients : {2,3}*12, {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  7)(  6,  8)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 73,161)( 74,162)( 75,164)( 76,163)( 77,167)( 78,168)( 79,165)
( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)( 86,160)( 87,157)
( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)( 93,151)( 94,152)( 95,149)
( 96,150)( 97,185)( 98,186)( 99,188)(100,187)(101,191)(102,192)(103,189)
(104,190)(105,177)(106,178)(107,180)(108,179)(109,183)(110,184)(111,181)
(112,182)(113,169)(114,170)(115,172)(116,171)(117,175)(118,176)(119,173)
(120,174)(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)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)(228,236)
(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)(249,258)
(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)(265,266)
(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)(278,287)
(279,286)(280,285)(289,378)(290,377)(291,379)(292,380)(293,384)(294,383)
(295,382)(296,381)(297,370)(298,369)(299,371)(300,372)(301,376)(302,375)
(303,374)(304,373)(305,362)(306,361)(307,363)(308,364)(309,368)(310,367)
(311,366)(312,365)(313,402)(314,401)(315,403)(316,404)(317,408)(318,407)
(319,406)(320,405)(321,394)(322,393)(323,395)(324,396)(325,400)(326,399)
(327,398)(328,397)(329,386)(330,385)(331,387)(332,388)(333,392)(334,391)
(335,390)(336,389)(337,426)(338,425)(339,427)(340,428)(341,432)(342,431)
(343,430)(344,429)(345,418)(346,417)(347,419)(348,420)(349,424)(350,423)
(351,422)(352,421)(353,410)(354,409)(355,411)(356,412)(357,416)(358,415)
(359,414)(360,413);;
s1 := (  1, 73)(  2, 74)(  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)(  8, 76)
(  9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)( 16, 92)
( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)( 24, 84)
( 25,121)( 26,122)( 27,127)( 28,128)( 29,126)( 30,125)( 31,123)( 32,124)
( 33,137)( 34,138)( 35,143)( 36,144)( 37,142)( 38,141)( 39,139)( 40,140)
( 41,129)( 42,130)( 43,135)( 44,136)( 45,134)( 46,133)( 47,131)( 48,132)
( 49, 97)( 50, 98)( 51,103)( 52,104)( 53,102)( 54,101)( 55, 99)( 56,100)
( 57,113)( 58,114)( 59,119)( 60,120)( 61,118)( 62,117)( 63,115)( 64,116)
( 65,105)( 66,106)( 67,111)( 68,112)( 69,110)( 70,109)( 71,107)( 72,108)
(145,161)(146,162)(147,167)(148,168)(149,166)(150,165)(151,163)(152,164)
(155,159)(156,160)(157,158)(169,209)(170,210)(171,215)(172,216)(173,214)
(174,213)(175,211)(176,212)(177,201)(178,202)(179,207)(180,208)(181,206)
(182,205)(183,203)(184,204)(185,193)(186,194)(187,199)(188,200)(189,198)
(190,197)(191,195)(192,196)(217,290)(218,289)(219,296)(220,295)(221,293)
(222,294)(223,292)(224,291)(225,306)(226,305)(227,312)(228,311)(229,309)
(230,310)(231,308)(232,307)(233,298)(234,297)(235,304)(236,303)(237,301)
(238,302)(239,300)(240,299)(241,338)(242,337)(243,344)(244,343)(245,341)
(246,342)(247,340)(248,339)(249,354)(250,353)(251,360)(252,359)(253,357)
(254,358)(255,356)(256,355)(257,346)(258,345)(259,352)(260,351)(261,349)
(262,350)(263,348)(264,347)(265,314)(266,313)(267,320)(268,319)(269,317)
(270,318)(271,316)(272,315)(273,330)(274,329)(275,336)(276,335)(277,333)
(278,334)(279,332)(280,331)(281,322)(282,321)(283,328)(284,327)(285,325)
(286,326)(287,324)(288,323)(361,378)(362,377)(363,384)(364,383)(365,381)
(366,382)(367,380)(368,379)(369,370)(371,376)(372,375)(385,426)(386,425)
(387,432)(388,431)(389,429)(390,430)(391,428)(392,427)(393,418)(394,417)
(395,424)(396,423)(397,421)(398,422)(399,420)(400,419)(401,410)(402,409)
(403,416)(404,415)(405,413)(406,414)(407,412)(408,411);;
s2 := (  1,243)(  2,244)(  3,242)(  4,241)(  5,247)(  6,248)(  7,246)(  8,245)
(  9,251)( 10,252)( 11,250)( 12,249)( 13,255)( 14,256)( 15,254)( 16,253)
( 17,259)( 18,260)( 19,258)( 20,257)( 21,263)( 22,264)( 23,262)( 24,261)
( 25,219)( 26,220)( 27,218)( 28,217)( 29,223)( 30,224)( 31,222)( 32,221)
( 33,227)( 34,228)( 35,226)( 36,225)( 37,231)( 38,232)( 39,230)( 40,229)
( 41,235)( 42,236)( 43,234)( 44,233)( 45,239)( 46,240)( 47,238)( 48,237)
( 49,267)( 50,268)( 51,266)( 52,265)( 53,271)( 54,272)( 55,270)( 56,269)
( 57,275)( 58,276)( 59,274)( 60,273)( 61,279)( 62,280)( 63,278)( 64,277)
( 65,283)( 66,284)( 67,282)( 68,281)( 69,287)( 70,288)( 71,286)( 72,285)
( 73,315)( 74,316)( 75,314)( 76,313)( 77,319)( 78,320)( 79,318)( 80,317)
( 81,323)( 82,324)( 83,322)( 84,321)( 85,327)( 86,328)( 87,326)( 88,325)
( 89,331)( 90,332)( 91,330)( 92,329)( 93,335)( 94,336)( 95,334)( 96,333)
( 97,291)( 98,292)( 99,290)(100,289)(101,295)(102,296)(103,294)(104,293)
(105,299)(106,300)(107,298)(108,297)(109,303)(110,304)(111,302)(112,301)
(113,307)(114,308)(115,306)(116,305)(117,311)(118,312)(119,310)(120,309)
(121,339)(122,340)(123,338)(124,337)(125,343)(126,344)(127,342)(128,341)
(129,347)(130,348)(131,346)(132,345)(133,351)(134,352)(135,350)(136,349)
(137,355)(138,356)(139,354)(140,353)(141,359)(142,360)(143,358)(144,357)
(145,387)(146,388)(147,386)(148,385)(149,391)(150,392)(151,390)(152,389)
(153,395)(154,396)(155,394)(156,393)(157,399)(158,400)(159,398)(160,397)
(161,403)(162,404)(163,402)(164,401)(165,407)(166,408)(167,406)(168,405)
(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)(176,365)
(177,371)(178,372)(179,370)(180,369)(181,375)(182,376)(183,374)(184,373)
(185,379)(186,380)(187,378)(188,377)(189,383)(190,384)(191,382)(192,381)
(193,411)(194,412)(195,410)(196,409)(197,415)(198,416)(199,414)(200,413)
(201,419)(202,420)(203,418)(204,417)(205,423)(206,424)(207,422)(208,421)
(209,427)(210,428)(211,426)(212,425)(213,431)(214,432)(215,430)(216,429);;
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*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!(  3,  4)(  5,  7)(  6,  8)(  9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 73,161)( 74,162)( 75,164)( 76,163)( 77,167)( 78,168)
( 79,165)( 80,166)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)( 86,160)
( 87,157)( 88,158)( 89,145)( 90,146)( 91,148)( 92,147)( 93,151)( 94,152)
( 95,149)( 96,150)( 97,185)( 98,186)( 99,188)(100,187)(101,191)(102,192)
(103,189)(104,190)(105,177)(106,178)(107,180)(108,179)(109,183)(110,184)
(111,181)(112,182)(113,169)(114,170)(115,172)(116,171)(117,175)(118,176)
(119,173)(120,174)(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)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)
(228,236)(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)
(249,258)(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)
(265,266)(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)
(278,287)(279,286)(280,285)(289,378)(290,377)(291,379)(292,380)(293,384)
(294,383)(295,382)(296,381)(297,370)(298,369)(299,371)(300,372)(301,376)
(302,375)(303,374)(304,373)(305,362)(306,361)(307,363)(308,364)(309,368)
(310,367)(311,366)(312,365)(313,402)(314,401)(315,403)(316,404)(317,408)
(318,407)(319,406)(320,405)(321,394)(322,393)(323,395)(324,396)(325,400)
(326,399)(327,398)(328,397)(329,386)(330,385)(331,387)(332,388)(333,392)
(334,391)(335,390)(336,389)(337,426)(338,425)(339,427)(340,428)(341,432)
(342,431)(343,430)(344,429)(345,418)(346,417)(347,419)(348,420)(349,424)
(350,423)(351,422)(352,421)(353,410)(354,409)(355,411)(356,412)(357,416)
(358,415)(359,414)(360,413);
s1 := Sym(432)!(  1, 73)(  2, 74)(  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)
(  8, 76)(  9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)
( 16, 92)( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)
( 24, 84)( 25,121)( 26,122)( 27,127)( 28,128)( 29,126)( 30,125)( 31,123)
( 32,124)( 33,137)( 34,138)( 35,143)( 36,144)( 37,142)( 38,141)( 39,139)
( 40,140)( 41,129)( 42,130)( 43,135)( 44,136)( 45,134)( 46,133)( 47,131)
( 48,132)( 49, 97)( 50, 98)( 51,103)( 52,104)( 53,102)( 54,101)( 55, 99)
( 56,100)( 57,113)( 58,114)( 59,119)( 60,120)( 61,118)( 62,117)( 63,115)
( 64,116)( 65,105)( 66,106)( 67,111)( 68,112)( 69,110)( 70,109)( 71,107)
( 72,108)(145,161)(146,162)(147,167)(148,168)(149,166)(150,165)(151,163)
(152,164)(155,159)(156,160)(157,158)(169,209)(170,210)(171,215)(172,216)
(173,214)(174,213)(175,211)(176,212)(177,201)(178,202)(179,207)(180,208)
(181,206)(182,205)(183,203)(184,204)(185,193)(186,194)(187,199)(188,200)
(189,198)(190,197)(191,195)(192,196)(217,290)(218,289)(219,296)(220,295)
(221,293)(222,294)(223,292)(224,291)(225,306)(226,305)(227,312)(228,311)
(229,309)(230,310)(231,308)(232,307)(233,298)(234,297)(235,304)(236,303)
(237,301)(238,302)(239,300)(240,299)(241,338)(242,337)(243,344)(244,343)
(245,341)(246,342)(247,340)(248,339)(249,354)(250,353)(251,360)(252,359)
(253,357)(254,358)(255,356)(256,355)(257,346)(258,345)(259,352)(260,351)
(261,349)(262,350)(263,348)(264,347)(265,314)(266,313)(267,320)(268,319)
(269,317)(270,318)(271,316)(272,315)(273,330)(274,329)(275,336)(276,335)
(277,333)(278,334)(279,332)(280,331)(281,322)(282,321)(283,328)(284,327)
(285,325)(286,326)(287,324)(288,323)(361,378)(362,377)(363,384)(364,383)
(365,381)(366,382)(367,380)(368,379)(369,370)(371,376)(372,375)(385,426)
(386,425)(387,432)(388,431)(389,429)(390,430)(391,428)(392,427)(393,418)
(394,417)(395,424)(396,423)(397,421)(398,422)(399,420)(400,419)(401,410)
(402,409)(403,416)(404,415)(405,413)(406,414)(407,412)(408,411);
s2 := Sym(432)!(  1,243)(  2,244)(  3,242)(  4,241)(  5,247)(  6,248)(  7,246)
(  8,245)(  9,251)( 10,252)( 11,250)( 12,249)( 13,255)( 14,256)( 15,254)
( 16,253)( 17,259)( 18,260)( 19,258)( 20,257)( 21,263)( 22,264)( 23,262)
( 24,261)( 25,219)( 26,220)( 27,218)( 28,217)( 29,223)( 30,224)( 31,222)
( 32,221)( 33,227)( 34,228)( 35,226)( 36,225)( 37,231)( 38,232)( 39,230)
( 40,229)( 41,235)( 42,236)( 43,234)( 44,233)( 45,239)( 46,240)( 47,238)
( 48,237)( 49,267)( 50,268)( 51,266)( 52,265)( 53,271)( 54,272)( 55,270)
( 56,269)( 57,275)( 58,276)( 59,274)( 60,273)( 61,279)( 62,280)( 63,278)
( 64,277)( 65,283)( 66,284)( 67,282)( 68,281)( 69,287)( 70,288)( 71,286)
( 72,285)( 73,315)( 74,316)( 75,314)( 76,313)( 77,319)( 78,320)( 79,318)
( 80,317)( 81,323)( 82,324)( 83,322)( 84,321)( 85,327)( 86,328)( 87,326)
( 88,325)( 89,331)( 90,332)( 91,330)( 92,329)( 93,335)( 94,336)( 95,334)
( 96,333)( 97,291)( 98,292)( 99,290)(100,289)(101,295)(102,296)(103,294)
(104,293)(105,299)(106,300)(107,298)(108,297)(109,303)(110,304)(111,302)
(112,301)(113,307)(114,308)(115,306)(116,305)(117,311)(118,312)(119,310)
(120,309)(121,339)(122,340)(123,338)(124,337)(125,343)(126,344)(127,342)
(128,341)(129,347)(130,348)(131,346)(132,345)(133,351)(134,352)(135,350)
(136,349)(137,355)(138,356)(139,354)(140,353)(141,359)(142,360)(143,358)
(144,357)(145,387)(146,388)(147,386)(148,385)(149,391)(150,392)(151,390)
(152,389)(153,395)(154,396)(155,394)(156,393)(157,399)(158,400)(159,398)
(160,397)(161,403)(162,404)(163,402)(164,401)(165,407)(166,408)(167,406)
(168,405)(169,363)(170,364)(171,362)(172,361)(173,367)(174,368)(175,366)
(176,365)(177,371)(178,372)(179,370)(180,369)(181,375)(182,376)(183,374)
(184,373)(185,379)(186,380)(187,378)(188,377)(189,383)(190,384)(191,382)
(192,381)(193,411)(194,412)(195,410)(196,409)(197,415)(198,416)(199,414)
(200,413)(201,419)(202,420)(203,418)(204,417)(205,423)(206,424)(207,422)
(208,421)(209,427)(210,428)(211,426)(212,425)(213,431)(214,432)(215,430)
(216,429);
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*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope