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# Polytope of Type {24,12}

Atlas Canonical Name : {24,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,5113)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
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 : {24,6}*864b, {12,12}*864c
3-fold quotients : {24,12}*576c
4-fold quotients : {6,12}*432b, {12,6}*432b
6-fold quotients : {24,6}*288a, {12,12}*288a
8-fold quotients : {6,6}*216b
9-fold quotients : {24,4}*192a, {8,12}*192a
12-fold quotients : {6,12}*144a, {12,6}*144a
16-fold quotients : {6,6}*108
18-fold quotients : {4,12}*96a, {12,4}*96a, {24,2}*96, {8,6}*96
24-fold quotients : {6,6}*72a
27-fold quotients : {8,4}*64a
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
54-fold quotients : {4,4}*32, {8,2}*32
72-fold quotients : {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
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 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)
(117,140)(118,145)(119,147)(120,146)(121,151)(122,153)(123,152)(124,148)
(125,150)(126,149)(127,154)(128,156)(129,155)(130,160)(131,162)(132,161)
(133,157)(134,159)(135,158)(163,190)(164,192)(165,191)(166,196)(167,198)
(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)(175,205)
(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)(183,209)
(184,214)(185,216)(186,215)(187,211)(188,213)(189,212)(217,325)(218,327)
(219,326)(220,331)(221,333)(222,332)(223,328)(224,330)(225,329)(226,334)
(227,336)(228,335)(229,340)(230,342)(231,341)(232,337)(233,339)(234,338)
(235,343)(236,345)(237,344)(238,349)(239,351)(240,350)(241,346)(242,348)
(243,347)(244,352)(245,354)(246,353)(247,358)(248,360)(249,359)(250,355)
(251,357)(252,356)(253,361)(254,363)(255,362)(256,367)(257,369)(258,368)
(259,364)(260,366)(261,365)(262,370)(263,372)(264,371)(265,376)(266,378)
(267,377)(268,373)(269,375)(270,374)(271,379)(272,381)(273,380)(274,385)
(275,387)(276,386)(277,382)(278,384)(279,383)(280,388)(281,390)(282,389)
(283,394)(284,396)(285,395)(286,391)(287,393)(288,392)(289,397)(290,399)
(291,398)(292,403)(293,405)(294,404)(295,400)(296,402)(297,401)(298,406)
(299,408)(300,407)(301,412)(302,414)(303,413)(304,409)(305,411)(306,410)
(307,415)(308,417)(309,416)(310,421)(311,423)(312,422)(313,418)(314,420)
(315,419)(316,424)(317,426)(318,425)(319,430)(320,432)(321,431)(322,427)
(323,429)(324,428);;
s1 := (  1,220)(  2,221)(  3,222)(  4,217)(  5,218)(  6,219)(  7,223)(  8,224)
(  9,225)( 10,238)( 11,239)( 12,240)( 13,235)( 14,236)( 15,237)( 16,241)
( 17,242)( 18,243)( 19,229)( 20,230)( 21,231)( 22,226)( 23,227)( 24,228)
( 25,232)( 26,233)( 27,234)( 28,247)( 29,248)( 30,249)( 31,244)( 32,245)
( 33,246)( 34,250)( 35,251)( 36,252)( 37,265)( 38,266)( 39,267)( 40,262)
( 41,263)( 42,264)( 43,268)( 44,269)( 45,270)( 46,256)( 47,257)( 48,258)
( 49,253)( 50,254)( 51,255)( 52,259)( 53,260)( 54,261)( 55,274)( 56,275)
( 57,276)( 58,271)( 59,272)( 60,273)( 61,277)( 62,278)( 63,279)( 64,292)
( 65,293)( 66,294)( 67,289)( 68,290)( 69,291)( 70,295)( 71,296)( 72,297)
( 73,283)( 74,284)( 75,285)( 76,280)( 77,281)( 78,282)( 79,286)( 80,287)
( 81,288)( 82,301)( 83,302)( 84,303)( 85,298)( 86,299)( 87,300)( 88,304)
( 89,305)( 90,306)( 91,319)( 92,320)( 93,321)( 94,316)( 95,317)( 96,318)
( 97,322)( 98,323)( 99,324)(100,310)(101,311)(102,312)(103,307)(104,308)
(105,309)(106,313)(107,314)(108,315)(109,355)(110,356)(111,357)(112,352)
(113,353)(114,354)(115,358)(116,359)(117,360)(118,373)(119,374)(120,375)
(121,370)(122,371)(123,372)(124,376)(125,377)(126,378)(127,364)(128,365)
(129,366)(130,361)(131,362)(132,363)(133,367)(134,368)(135,369)(136,328)
(137,329)(138,330)(139,325)(140,326)(141,327)(142,331)(143,332)(144,333)
(145,346)(146,347)(147,348)(148,343)(149,344)(150,345)(151,349)(152,350)
(153,351)(154,337)(155,338)(156,339)(157,334)(158,335)(159,336)(160,340)
(161,341)(162,342)(163,409)(164,410)(165,411)(166,406)(167,407)(168,408)
(169,412)(170,413)(171,414)(172,427)(173,428)(174,429)(175,424)(176,425)
(177,426)(178,430)(179,431)(180,432)(181,418)(182,419)(183,420)(184,415)
(185,416)(186,417)(187,421)(188,422)(189,423)(190,382)(191,383)(192,384)
(193,379)(194,380)(195,381)(196,385)(197,386)(198,387)(199,400)(200,401)
(201,402)(202,397)(203,398)(204,399)(205,403)(206,404)(207,405)(208,391)
(209,392)(210,393)(211,388)(212,389)(213,390)(214,394)(215,395)(216,396);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)(  8, 17)
(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)
( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)( 62, 71)
( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 95)
( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)(106,108)
(109,118)(110,120)(111,119)(112,122)(113,121)(114,123)(115,126)(116,125)
(117,124)(128,129)(130,131)(133,135)(136,145)(137,147)(138,146)(139,149)
(140,148)(141,150)(142,153)(143,152)(144,151)(155,156)(157,158)(160,162)
(163,172)(164,174)(165,173)(166,176)(167,175)(168,177)(169,180)(170,179)
(171,178)(182,183)(184,185)(187,189)(190,199)(191,201)(192,200)(193,203)
(194,202)(195,204)(196,207)(197,206)(198,205)(209,210)(211,212)(214,216)
(217,280)(218,282)(219,281)(220,284)(221,283)(222,285)(223,288)(224,287)
(225,286)(226,271)(227,273)(228,272)(229,275)(230,274)(231,276)(232,279)
(233,278)(234,277)(235,289)(236,291)(237,290)(238,293)(239,292)(240,294)
(241,297)(242,296)(243,295)(244,307)(245,309)(246,308)(247,311)(248,310)
(249,312)(250,315)(251,314)(252,313)(253,298)(254,300)(255,299)(256,302)
(257,301)(258,303)(259,306)(260,305)(261,304)(262,316)(263,318)(264,317)
(265,320)(266,319)(267,321)(268,324)(269,323)(270,322)(325,388)(326,390)
(327,389)(328,392)(329,391)(330,393)(331,396)(332,395)(333,394)(334,379)
(335,381)(336,380)(337,383)(338,382)(339,384)(340,387)(341,386)(342,385)
(343,397)(344,399)(345,398)(346,401)(347,400)(348,402)(349,405)(350,404)
(351,403)(352,415)(353,417)(354,416)(355,419)(356,418)(357,420)(358,423)
(359,422)(360,421)(361,406)(362,408)(363,407)(364,410)(365,409)(366,411)
(367,414)(368,413)(369,412)(370,424)(371,426)(372,425)(373,428)(374,427)
(375,429)(376,432)(377,431)(378,430);;
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*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*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,
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope