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# Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728l
if this polytope has a name.
Group : SmallGroup(1728,30242)
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
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 : {12,6}*864b
3-fold quotients : {12,12}*576f
4-fold quotients : {6,12}*432b, {12,6}*432d
6-fold quotients : {12,6}*288a
8-fold quotients : {6,6}*216b
9-fold quotients : {4,12}*192b
12-fold quotients : {6,12}*144a, {12,6}*144d
16-fold quotients : {6,6}*108
18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
24-fold quotients : {6,6}*72a
36-fold quotients : {2,12}*48, {6,4}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
72-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*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,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)
( 23, 29)( 24, 30)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 71)( 54, 72)( 55, 69)( 56, 70)
( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 73, 75)( 74, 76)( 77, 83)( 78, 84)
( 79, 81)( 80, 82)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,107)( 90,108)
( 91,105)( 92,106)( 93,103)( 94,104)( 95,101)( 96,102)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,135)(122,136)(123,133)(124,134)
(125,143)(126,144)(127,141)(128,142)(129,139)(130,140)(131,137)(132,138)
(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,171)(158,172)
(159,169)(160,170)(161,179)(162,180)(163,177)(164,178)(165,175)(166,176)
(167,173)(168,174)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,207)(194,208)(195,205)(196,206)(197,215)(198,216)(199,213)(200,214)
(201,211)(202,212)(203,209)(204,210)(217,219)(218,220)(221,227)(222,228)
(223,225)(224,226)(229,243)(230,244)(231,241)(232,242)(233,251)(234,252)
(235,249)(236,250)(237,247)(238,248)(239,245)(240,246)(253,255)(254,256)
(257,263)(258,264)(259,261)(260,262)(265,279)(266,280)(267,277)(268,278)
(269,287)(270,288)(271,285)(272,286)(273,283)(274,284)(275,281)(276,282)
(289,291)(290,292)(293,299)(294,300)(295,297)(296,298)(301,315)(302,316)
(303,313)(304,314)(305,323)(306,324)(307,321)(308,322)(309,319)(310,320)
(311,317)(312,318)(325,327)(326,328)(329,335)(330,336)(331,333)(332,334)
(337,351)(338,352)(339,349)(340,350)(341,359)(342,360)(343,357)(344,358)
(345,355)(346,356)(347,353)(348,354)(361,363)(362,364)(365,371)(366,372)
(367,369)(368,370)(373,387)(374,388)(375,385)(376,386)(377,395)(378,396)
(379,393)(380,394)(381,391)(382,392)(383,389)(384,390)(397,399)(398,400)
(401,407)(402,408)(403,405)(404,406)(409,423)(410,424)(411,421)(412,422)
(413,431)(414,432)(415,429)(416,430)(417,427)(418,428)(419,425)(420,426);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)(  8, 19)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)( 37, 85)
( 38, 86)( 39, 88)( 40, 87)( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 93)
( 46, 94)( 47, 96)( 48, 95)( 49, 73)( 50, 74)( 51, 76)( 52, 75)( 53, 77)
( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 82)( 59, 84)( 60, 83)( 61, 97)
( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)
( 70,106)( 71,108)( 72,107)(109,121)(110,122)(111,124)(112,123)(113,125)
(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)(120,131)(135,136)
(139,140)(143,144)(145,193)(146,194)(147,196)(148,195)(149,197)(150,198)
(151,200)(152,199)(153,201)(154,202)(155,204)(156,203)(157,181)(158,182)
(159,184)(160,183)(161,185)(162,186)(163,188)(164,187)(165,189)(166,190)
(167,192)(168,191)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)
(175,212)(176,211)(177,213)(178,214)(179,216)(180,215)(217,337)(218,338)
(219,340)(220,339)(221,341)(222,342)(223,344)(224,343)(225,345)(226,346)
(227,348)(228,347)(229,325)(230,326)(231,328)(232,327)(233,329)(234,330)
(235,332)(236,331)(237,333)(238,334)(239,336)(240,335)(241,349)(242,350)
(243,352)(244,351)(245,353)(246,354)(247,356)(248,355)(249,357)(250,358)
(251,360)(252,359)(253,409)(254,410)(255,412)(256,411)(257,413)(258,414)
(259,416)(260,415)(261,417)(262,418)(263,420)(264,419)(265,397)(266,398)
(267,400)(268,399)(269,401)(270,402)(271,404)(272,403)(273,405)(274,406)
(275,408)(276,407)(277,421)(278,422)(279,424)(280,423)(281,425)(282,426)
(283,428)(284,427)(285,429)(286,430)(287,432)(288,431)(289,373)(290,374)
(291,376)(292,375)(293,377)(294,378)(295,380)(296,379)(297,381)(298,382)
(299,384)(300,383)(301,361)(302,362)(303,364)(304,363)(305,365)(306,366)
(307,368)(308,367)(309,369)(310,370)(311,372)(312,371)(313,385)(314,386)
(315,388)(316,387)(317,389)(318,390)(319,392)(320,391)(321,393)(322,394)
(323,396)(324,395);;
s2 := (  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)(  8,262)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,269)( 14,272)( 15,271)( 16,270)
( 17,265)( 18,268)( 19,267)( 20,266)( 21,273)( 22,276)( 23,275)( 24,274)
( 25,285)( 26,288)( 27,287)( 28,286)( 29,281)( 30,284)( 31,283)( 32,282)
( 33,277)( 34,280)( 35,279)( 36,278)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,233)( 50,236)( 51,235)( 52,234)( 53,229)( 54,232)( 55,231)( 56,230)
( 57,237)( 58,240)( 59,239)( 60,238)( 61,249)( 62,252)( 63,251)( 64,250)
( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)( 72,242)
( 73,289)( 74,292)( 75,291)( 76,290)( 77,297)( 78,300)( 79,299)( 80,298)
( 81,293)( 82,296)( 83,295)( 84,294)( 85,305)( 86,308)( 87,307)( 88,306)
( 89,301)( 90,304)( 91,303)( 92,302)( 93,309)( 94,312)( 95,311)( 96,310)
( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)(104,318)
(105,313)(106,316)(107,315)(108,314)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,377)(122,380)(123,379)(124,378)(125,373)(126,376)(127,375)(128,374)
(129,381)(130,384)(131,383)(132,382)(133,393)(134,396)(135,395)(136,394)
(137,389)(138,392)(139,391)(140,390)(141,385)(142,388)(143,387)(144,386)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,341)(158,344)(159,343)(160,342)
(161,337)(162,340)(163,339)(164,338)(165,345)(166,348)(167,347)(168,346)
(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)
(177,349)(178,352)(179,351)(180,350)(181,397)(182,400)(183,399)(184,398)
(185,405)(186,408)(187,407)(188,406)(189,401)(190,404)(191,403)(192,402)
(193,413)(194,416)(195,415)(196,414)(197,409)(198,412)(199,411)(200,410)
(201,417)(202,420)(203,419)(204,418)(205,429)(206,432)(207,431)(208,430)
(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)(216,422);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*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,
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,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)
( 22, 32)( 23, 29)( 24, 30)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)
( 44, 46)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 71)( 54, 72)( 55, 69)
( 56, 70)( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 73, 75)( 74, 76)( 77, 83)
( 78, 84)( 79, 81)( 80, 82)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,107)
( 90,108)( 91,105)( 92,106)( 93,103)( 94,104)( 95,101)( 96,102)(109,111)
(110,112)(113,119)(114,120)(115,117)(116,118)(121,135)(122,136)(123,133)
(124,134)(125,143)(126,144)(127,141)(128,142)(129,139)(130,140)(131,137)
(132,138)(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,171)
(158,172)(159,169)(160,170)(161,179)(162,180)(163,177)(164,178)(165,175)
(166,176)(167,173)(168,174)(181,183)(182,184)(185,191)(186,192)(187,189)
(188,190)(193,207)(194,208)(195,205)(196,206)(197,215)(198,216)(199,213)
(200,214)(201,211)(202,212)(203,209)(204,210)(217,219)(218,220)(221,227)
(222,228)(223,225)(224,226)(229,243)(230,244)(231,241)(232,242)(233,251)
(234,252)(235,249)(236,250)(237,247)(238,248)(239,245)(240,246)(253,255)
(254,256)(257,263)(258,264)(259,261)(260,262)(265,279)(266,280)(267,277)
(268,278)(269,287)(270,288)(271,285)(272,286)(273,283)(274,284)(275,281)
(276,282)(289,291)(290,292)(293,299)(294,300)(295,297)(296,298)(301,315)
(302,316)(303,313)(304,314)(305,323)(306,324)(307,321)(308,322)(309,319)
(310,320)(311,317)(312,318)(325,327)(326,328)(329,335)(330,336)(331,333)
(332,334)(337,351)(338,352)(339,349)(340,350)(341,359)(342,360)(343,357)
(344,358)(345,355)(346,356)(347,353)(348,354)(361,363)(362,364)(365,371)
(366,372)(367,369)(368,370)(373,387)(374,388)(375,385)(376,386)(377,395)
(378,396)(379,393)(380,394)(381,391)(382,392)(383,389)(384,390)(397,399)
(398,400)(401,407)(402,408)(403,405)(404,406)(409,423)(410,424)(411,421)
(412,422)(413,431)(414,432)(415,429)(416,430)(417,427)(418,428)(419,425)
(420,426);
s1 := Sym(432)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)
(  8, 19)(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)
( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 89)( 42, 90)( 43, 92)( 44, 91)
( 45, 93)( 46, 94)( 47, 96)( 48, 95)( 49, 73)( 50, 74)( 51, 76)( 52, 75)
( 53, 77)( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 82)( 59, 84)( 60, 83)
( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)
( 69,105)( 70,106)( 71,108)( 72,107)(109,121)(110,122)(111,124)(112,123)
(113,125)(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)(120,131)
(135,136)(139,140)(143,144)(145,193)(146,194)(147,196)(148,195)(149,197)
(150,198)(151,200)(152,199)(153,201)(154,202)(155,204)(156,203)(157,181)
(158,182)(159,184)(160,183)(161,185)(162,186)(163,188)(164,187)(165,189)
(166,190)(167,192)(168,191)(169,205)(170,206)(171,208)(172,207)(173,209)
(174,210)(175,212)(176,211)(177,213)(178,214)(179,216)(180,215)(217,337)
(218,338)(219,340)(220,339)(221,341)(222,342)(223,344)(224,343)(225,345)
(226,346)(227,348)(228,347)(229,325)(230,326)(231,328)(232,327)(233,329)
(234,330)(235,332)(236,331)(237,333)(238,334)(239,336)(240,335)(241,349)
(242,350)(243,352)(244,351)(245,353)(246,354)(247,356)(248,355)(249,357)
(250,358)(251,360)(252,359)(253,409)(254,410)(255,412)(256,411)(257,413)
(258,414)(259,416)(260,415)(261,417)(262,418)(263,420)(264,419)(265,397)
(266,398)(267,400)(268,399)(269,401)(270,402)(271,404)(272,403)(273,405)
(274,406)(275,408)(276,407)(277,421)(278,422)(279,424)(280,423)(281,425)
(282,426)(283,428)(284,427)(285,429)(286,430)(287,432)(288,431)(289,373)
(290,374)(291,376)(292,375)(293,377)(294,378)(295,380)(296,379)(297,381)
(298,382)(299,384)(300,383)(301,361)(302,362)(303,364)(304,363)(305,365)
(306,366)(307,368)(308,367)(309,369)(310,370)(311,372)(312,371)(313,385)
(314,386)(315,388)(316,387)(317,389)(318,390)(319,392)(320,391)(321,393)
(322,394)(323,396)(324,395);
s2 := Sym(432)!(  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)
(  8,262)(  9,257)( 10,260)( 11,259)( 12,258)( 13,269)( 14,272)( 15,271)
( 16,270)( 17,265)( 18,268)( 19,267)( 20,266)( 21,273)( 22,276)( 23,275)
( 24,274)( 25,285)( 26,288)( 27,287)( 28,286)( 29,281)( 30,284)( 31,283)
( 32,282)( 33,277)( 34,280)( 35,279)( 36,278)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,233)( 50,236)( 51,235)( 52,234)( 53,229)( 54,232)( 55,231)
( 56,230)( 57,237)( 58,240)( 59,239)( 60,238)( 61,249)( 62,252)( 63,251)
( 64,250)( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)
( 72,242)( 73,289)( 74,292)( 75,291)( 76,290)( 77,297)( 78,300)( 79,299)
( 80,298)( 81,293)( 82,296)( 83,295)( 84,294)( 85,305)( 86,308)( 87,307)
( 88,306)( 89,301)( 90,304)( 91,303)( 92,302)( 93,309)( 94,312)( 95,311)
( 96,310)( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)
(104,318)(105,313)(106,316)(107,315)(108,314)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,377)(122,380)(123,379)(124,378)(125,373)(126,376)(127,375)
(128,374)(129,381)(130,384)(131,383)(132,382)(133,393)(134,396)(135,395)
(136,394)(137,389)(138,392)(139,391)(140,390)(141,385)(142,388)(143,387)
(144,386)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,341)(158,344)(159,343)
(160,342)(161,337)(162,340)(163,339)(164,338)(165,345)(166,348)(167,347)
(168,346)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)
(176,354)(177,349)(178,352)(179,351)(180,350)(181,397)(182,400)(183,399)
(184,398)(185,405)(186,408)(187,407)(188,406)(189,401)(190,404)(191,403)
(192,402)(193,413)(194,416)(195,415)(196,414)(197,409)(198,412)(199,411)
(200,410)(201,417)(202,420)(203,419)(204,418)(205,429)(206,432)(207,431)
(208,430)(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)
(216,422);
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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*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,
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