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# Polytope of Type {12,24}

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

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

```
References : None.
to this polytope