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# Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30326)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 72, 432, 144
Order of s0s1s2 : 12
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 : {6,6}*864a
3-fold quotients : {6,12}*576e
4-fold quotients : {6,12}*432c, {3,6}*432
6-fold quotients : {6,6}*288b
8-fold quotients : {6,6}*216c
9-fold quotients : {6,12}*192a
12-fold quotients : {6,12}*144c, {3,6}*144
16-fold quotients : {3,6}*108
18-fold quotients : {6,6}*96
24-fold quotients : {6,6}*72c
36-fold quotients : {6,4}*48a, {3,6}*48, {6,3}*48
48-fold quotients : {3,6}*36
72-fold quotients : {3,3}*24, {6,2}*24
108-fold quotients : {2,4}*16
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)
( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)
( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)
( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)( 61, 85)
( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)
( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)(122,134)
(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)
(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)
(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)(158,206)
(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)(166,214)
(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)(174,198)
(175,200)(176,199)(177,201)(178,202)(179,204)(180,203)(219,220)(223,224)
(227,228)(229,241)(230,242)(231,244)(232,243)(233,245)(234,246)(235,248)
(236,247)(237,249)(238,250)(239,252)(240,251)(253,289)(254,290)(255,292)
(256,291)(257,293)(258,294)(259,296)(260,295)(261,297)(262,298)(263,300)
(264,299)(265,313)(266,314)(267,316)(268,315)(269,317)(270,318)(271,320)
(272,319)(273,321)(274,322)(275,324)(276,323)(277,301)(278,302)(279,304)
(280,303)(281,305)(282,306)(283,308)(284,307)(285,309)(286,310)(287,312)
(288,311)(327,328)(331,332)(335,336)(337,349)(338,350)(339,352)(340,351)
(341,353)(342,354)(343,356)(344,355)(345,357)(346,358)(347,360)(348,359)
(361,397)(362,398)(363,400)(364,399)(365,401)(366,402)(367,404)(368,403)
(369,405)(370,406)(371,408)(372,407)(373,421)(374,422)(375,424)(376,423)
(377,425)(378,426)(379,428)(380,427)(381,429)(382,430)(383,432)(384,431)
(385,409)(386,410)(387,412)(388,411)(389,413)(390,414)(391,416)(392,415)
(393,417)(394,418)(395,420)(396,419);;
s1 := (  1,305)(  2,307)(  3,306)(  4,308)(  5,309)(  6,311)(  7,310)(  8,312)
(  9,301)( 10,303)( 11,302)( 12,304)( 13,297)( 14,299)( 15,298)( 16,300)
( 17,289)( 18,291)( 19,290)( 20,292)( 21,293)( 22,295)( 23,294)( 24,296)
( 25,313)( 26,315)( 27,314)( 28,316)( 29,317)( 30,319)( 31,318)( 32,320)
( 33,321)( 34,323)( 35,322)( 36,324)( 37,269)( 38,271)( 39,270)( 40,272)
( 41,273)( 42,275)( 43,274)( 44,276)( 45,265)( 46,267)( 47,266)( 48,268)
( 49,261)( 50,263)( 51,262)( 52,264)( 53,253)( 54,255)( 55,254)( 56,256)
( 57,257)( 58,259)( 59,258)( 60,260)( 61,277)( 62,279)( 63,278)( 64,280)
( 65,281)( 66,283)( 67,282)( 68,284)( 69,285)( 70,287)( 71,286)( 72,288)
( 73,233)( 74,235)( 75,234)( 76,236)( 77,237)( 78,239)( 79,238)( 80,240)
( 81,229)( 82,231)( 83,230)( 84,232)( 85,225)( 86,227)( 87,226)( 88,228)
( 89,217)( 90,219)( 91,218)( 92,220)( 93,221)( 94,223)( 95,222)( 96,224)
( 97,241)( 98,243)( 99,242)(100,244)(101,245)(102,247)(103,246)(104,248)
(105,249)(106,251)(107,250)(108,252)(109,413)(110,415)(111,414)(112,416)
(113,417)(114,419)(115,418)(116,420)(117,409)(118,411)(119,410)(120,412)
(121,405)(122,407)(123,406)(124,408)(125,397)(126,399)(127,398)(128,400)
(129,401)(130,403)(131,402)(132,404)(133,421)(134,423)(135,422)(136,424)
(137,425)(138,427)(139,426)(140,428)(141,429)(142,431)(143,430)(144,432)
(145,377)(146,379)(147,378)(148,380)(149,381)(150,383)(151,382)(152,384)
(153,373)(154,375)(155,374)(156,376)(157,369)(158,371)(159,370)(160,372)
(161,361)(162,363)(163,362)(164,364)(165,365)(166,367)(167,366)(168,368)
(169,385)(170,387)(171,386)(172,388)(173,389)(174,391)(175,390)(176,392)
(177,393)(178,395)(179,394)(180,396)(181,341)(182,343)(183,342)(184,344)
(185,345)(186,347)(187,346)(188,348)(189,337)(190,339)(191,338)(192,340)
(193,333)(194,335)(195,334)(196,336)(197,325)(198,327)(199,326)(200,328)
(201,329)(202,331)(203,330)(204,332)(205,349)(206,351)(207,350)(208,352)
(209,353)(210,355)(211,354)(212,356)(213,357)(214,359)(215,358)(216,360);;
s2 := (  1,  2)(  5, 10)(  6,  9)(  7, 11)(  8, 12)( 13, 14)( 17, 22)( 18, 21)
( 19, 23)( 20, 24)( 25, 26)( 29, 34)( 30, 33)( 31, 35)( 32, 36)( 37, 74)
( 38, 73)( 39, 75)( 40, 76)( 41, 82)( 42, 81)( 43, 83)( 44, 84)( 45, 78)
( 46, 77)( 47, 79)( 48, 80)( 49, 86)( 50, 85)( 51, 87)( 52, 88)( 53, 94)
( 54, 93)( 55, 95)( 56, 96)( 57, 90)( 58, 89)( 59, 91)( 60, 92)( 61, 98)
( 62, 97)( 63, 99)( 64,100)( 65,106)( 66,105)( 67,107)( 68,108)( 69,102)
( 70,101)( 71,103)( 72,104)(109,110)(113,118)(114,117)(115,119)(116,120)
(121,122)(125,130)(126,129)(127,131)(128,132)(133,134)(137,142)(138,141)
(139,143)(140,144)(145,182)(146,181)(147,183)(148,184)(149,190)(150,189)
(151,191)(152,192)(153,186)(154,185)(155,187)(156,188)(157,194)(158,193)
(159,195)(160,196)(161,202)(162,201)(163,203)(164,204)(165,198)(166,197)
(167,199)(168,200)(169,206)(170,205)(171,207)(172,208)(173,214)(174,213)
(175,215)(176,216)(177,210)(178,209)(179,211)(180,212)(217,326)(218,325)
(219,327)(220,328)(221,334)(222,333)(223,335)(224,336)(225,330)(226,329)
(227,331)(228,332)(229,338)(230,337)(231,339)(232,340)(233,346)(234,345)
(235,347)(236,348)(237,342)(238,341)(239,343)(240,344)(241,350)(242,349)
(243,351)(244,352)(245,358)(246,357)(247,359)(248,360)(249,354)(250,353)
(251,355)(252,356)(253,398)(254,397)(255,399)(256,400)(257,406)(258,405)
(259,407)(260,408)(261,402)(262,401)(263,403)(264,404)(265,410)(266,409)
(267,411)(268,412)(269,418)(270,417)(271,419)(272,420)(273,414)(274,413)
(275,415)(276,416)(277,422)(278,421)(279,423)(280,424)(281,430)(282,429)
(283,431)(284,432)(285,426)(286,425)(287,427)(288,428)(289,362)(290,361)
(291,363)(292,364)(293,370)(294,369)(295,371)(296,372)(297,366)(298,365)
(299,367)(300,368)(301,374)(302,373)(303,375)(304,376)(305,382)(306,381)
(307,383)(308,384)(309,378)(310,377)(311,379)(312,380)(313,386)(314,385)
(315,387)(316,388)(317,394)(318,393)(319,395)(320,396)(321,390)(322,389)
(323,391)(324,392);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope