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# Polytope of Type {12,12}

Atlas Canonical Name : {12,12}*1728h
if this polytope has a name.
Group : SmallGroup(1728,23054)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Dual
Self-Petrie
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 : {12,12}*864h
3-fold quotients : {12,12}*576a, {12,12}*576b, {12,12}*576c
4-fold quotients : {6,12}*432g, {12,6}*432g
6-fold quotients : {12,12}*288a, {12,12}*288b, {12,12}*288c
8-fold quotients : {6,6}*216d
9-fold quotients : {4,12}*192a, {12,4}*192a
12-fold quotients : {6,12}*144a, {6,12}*144b, {12,6}*144a, {12,6}*144b, {6,12}*144c, {12,6}*144c
18-fold quotients : {4,12}*96a, {12,4}*96a
24-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
27-fold quotients : {4,4}*64
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
48-fold quotients : {3,6}*36, {6,3}*36
54-fold quotients : {4,4}*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 := (  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)
(  9,113)( 10,118)( 11,120)( 12,119)( 13,124)( 14,126)( 15,125)( 16,121)
( 17,123)( 18,122)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)( 24,134)
( 25,130)( 26,132)( 27,131)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)
( 33,143)( 34,139)( 35,141)( 36,140)( 37,145)( 38,147)( 39,146)( 40,151)
( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,154)( 47,156)( 48,155)
( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,190)( 56,192)
( 57,191)( 58,196)( 59,198)( 60,197)( 61,193)( 62,195)( 63,194)( 64,199)
( 65,201)( 66,200)( 67,205)( 68,207)( 69,206)( 70,202)( 71,204)( 72,203)
( 73,208)( 74,210)( 75,209)( 76,214)( 77,216)( 78,215)( 79,211)( 80,213)
( 81,212)( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)
( 89,168)( 90,167)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)
( 97,175)( 98,177)( 99,176)(100,181)(101,183)(102,182)(103,187)(104,189)
(105,188)(106,184)(107,186)(108,185)(217,379)(218,381)(219,380)(220,385)
(221,387)(222,386)(223,382)(224,384)(225,383)(226,388)(227,390)(228,389)
(229,394)(230,396)(231,395)(232,391)(233,393)(234,392)(235,397)(236,399)
(237,398)(238,403)(239,405)(240,404)(241,400)(242,402)(243,401)(244,406)
(245,408)(246,407)(247,412)(248,414)(249,413)(250,409)(251,411)(252,410)
(253,415)(254,417)(255,416)(256,421)(257,423)(258,422)(259,418)(260,420)
(261,419)(262,424)(263,426)(264,425)(265,430)(266,432)(267,431)(268,427)
(269,429)(270,428)(271,352)(272,354)(273,353)(274,358)(275,360)(276,359)
(277,355)(278,357)(279,356)(280,361)(281,363)(282,362)(283,367)(284,369)
(285,368)(286,364)(287,366)(288,365)(289,370)(290,372)(291,371)(292,376)
(293,378)(294,377)(295,373)(296,375)(297,374)(298,325)(299,327)(300,326)
(301,331)(302,333)(303,332)(304,328)(305,330)(306,329)(307,334)(308,336)
(309,335)(310,340)(311,342)(312,341)(313,337)(314,339)(315,338)(316,343)
(317,345)(318,344)(319,349)(320,351)(321,350)(322,346)(323,348)(324,347);;
s1 := (  1,329)(  2,328)(  3,330)(  4,326)(  5,325)(  6,327)(  7,332)(  8,331)
(  9,333)( 10,347)( 11,346)( 12,348)( 13,344)( 14,343)( 15,345)( 16,350)
( 17,349)( 18,351)( 19,338)( 20,337)( 21,339)( 22,335)( 23,334)( 24,336)
( 25,341)( 26,340)( 27,342)( 28,356)( 29,355)( 30,357)( 31,353)( 32,352)
( 33,354)( 34,359)( 35,358)( 36,360)( 37,374)( 38,373)( 39,375)( 40,371)
( 41,370)( 42,372)( 43,377)( 44,376)( 45,378)( 46,365)( 47,364)( 48,366)
( 49,362)( 50,361)( 51,363)( 52,368)( 53,367)( 54,369)( 55,383)( 56,382)
( 57,384)( 58,380)( 59,379)( 60,381)( 61,386)( 62,385)( 63,387)( 64,401)
( 65,400)( 66,402)( 67,398)( 68,397)( 69,399)( 70,404)( 71,403)( 72,405)
( 73,392)( 74,391)( 75,393)( 76,389)( 77,388)( 78,390)( 79,395)( 80,394)
( 81,396)( 82,410)( 83,409)( 84,411)( 85,407)( 86,406)( 87,408)( 88,413)
( 89,412)( 90,414)( 91,428)( 92,427)( 93,429)( 94,425)( 95,424)( 96,426)
( 97,431)( 98,430)( 99,432)(100,419)(101,418)(102,420)(103,416)(104,415)
(105,417)(106,422)(107,421)(108,423)(109,248)(110,247)(111,249)(112,245)
(113,244)(114,246)(115,251)(116,250)(117,252)(118,266)(119,265)(120,267)
(121,263)(122,262)(123,264)(124,269)(125,268)(126,270)(127,257)(128,256)
(129,258)(130,254)(131,253)(132,255)(133,260)(134,259)(135,261)(136,221)
(137,220)(138,222)(139,218)(140,217)(141,219)(142,224)(143,223)(144,225)
(145,239)(146,238)(147,240)(148,236)(149,235)(150,237)(151,242)(152,241)
(153,243)(154,230)(155,229)(156,231)(157,227)(158,226)(159,228)(160,233)
(161,232)(162,234)(163,302)(164,301)(165,303)(166,299)(167,298)(168,300)
(169,305)(170,304)(171,306)(172,320)(173,319)(174,321)(175,317)(176,316)
(177,318)(178,323)(179,322)(180,324)(181,311)(182,310)(183,312)(184,308)
(185,307)(186,309)(187,314)(188,313)(189,315)(190,275)(191,274)(192,276)
(193,272)(194,271)(195,273)(196,278)(197,277)(198,279)(199,293)(200,292)
(201,294)(202,290)(203,289)(204,291)(205,296)(206,295)(207,297)(208,284)
(209,283)(210,285)(211,281)(212,280)(213,282)(214,287)(215,286)(216,288);;
s2 := (  1,253)(  2,255)(  3,254)(  4,256)(  5,258)(  6,257)(  7,259)(  8,261)
(  9,260)( 10,244)( 11,246)( 12,245)( 13,247)( 14,249)( 15,248)( 16,250)
( 17,252)( 18,251)( 19,262)( 20,264)( 21,263)( 22,265)( 23,267)( 24,266)
( 25,268)( 26,270)( 27,269)( 28,226)( 29,228)( 30,227)( 31,229)( 32,231)
( 33,230)( 34,232)( 35,234)( 36,233)( 37,217)( 38,219)( 39,218)( 40,220)
( 41,222)( 42,221)( 43,223)( 44,225)( 45,224)( 46,235)( 47,237)( 48,236)
( 49,238)( 50,240)( 51,239)( 52,241)( 53,243)( 54,242)( 55,307)( 56,309)
( 57,308)( 58,310)( 59,312)( 60,311)( 61,313)( 62,315)( 63,314)( 64,298)
( 65,300)( 66,299)( 67,301)( 68,303)( 69,302)( 70,304)( 71,306)( 72,305)
( 73,316)( 74,318)( 75,317)( 76,319)( 77,321)( 78,320)( 79,322)( 80,324)
( 81,323)( 82,280)( 83,282)( 84,281)( 85,283)( 86,285)( 87,284)( 88,286)
( 89,288)( 90,287)( 91,271)( 92,273)( 93,272)( 94,274)( 95,276)( 96,275)
( 97,277)( 98,279)( 99,278)(100,289)(101,291)(102,290)(103,292)(104,294)
(105,293)(106,295)(107,297)(108,296)(109,415)(110,417)(111,416)(112,418)
(113,420)(114,419)(115,421)(116,423)(117,422)(118,406)(119,408)(120,407)
(121,409)(122,411)(123,410)(124,412)(125,414)(126,413)(127,424)(128,426)
(129,425)(130,427)(131,429)(132,428)(133,430)(134,432)(135,431)(136,388)
(137,390)(138,389)(139,391)(140,393)(141,392)(142,394)(143,396)(144,395)
(145,379)(146,381)(147,380)(148,382)(149,384)(150,383)(151,385)(152,387)
(153,386)(154,397)(155,399)(156,398)(157,400)(158,402)(159,401)(160,403)
(161,405)(162,404)(163,361)(164,363)(165,362)(166,364)(167,366)(168,365)
(169,367)(170,369)(171,368)(172,352)(173,354)(174,353)(175,355)(176,357)
(177,356)(178,358)(179,360)(180,359)(181,370)(182,372)(183,371)(184,373)
(185,375)(186,374)(187,376)(188,378)(189,377)(190,334)(191,336)(192,335)
(193,337)(194,339)(195,338)(196,340)(197,342)(198,341)(199,325)(200,327)
(201,326)(202,328)(203,330)(204,329)(205,331)(206,333)(207,332)(208,343)
(209,345)(210,344)(211,346)(212,348)(213,347)(214,349)(215,351)(216,350);;
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*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope