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# Polytope of Type {72,12}

Atlas Canonical Name : {72,12}*1728a
Also Known As : {72,12|2}. if this polytope has another name.
Group : SmallGroup(1728,4714)
Rank : 3
Schlafli Type : {72,12}
Number of vertices, edges, etc : 72, 432, 12
Order of s0s1s2 : 72
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {72,6}*864a, {36,12}*864a
3-fold quotients : {72,4}*576a, {24,12}*576c
4-fold quotients : {36,6}*432a, {18,12}*432a
6-fold quotients : {36,4}*288a, {72,2}*288, {24,6}*288a, {12,12}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {24,4}*192a, {8,12}*192a
12-fold quotients : {36,2}*144, {18,4}*144a, {6,12}*144a, {12,6}*144a
18-fold quotients : {4,12}*96a, {12,4}*96a, {24,2}*96, {8,6}*96
24-fold quotients : {18,2}*72, {6,6}*72a
27-fold quotients : {8,4}*64a
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
48-fold quotients : {9,2}*36
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,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)
( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)( 33, 34)
( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)( 51, 52)
( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 65, 66)( 67, 72)( 68, 71)( 69, 70)
( 74, 75)( 76, 81)( 77, 80)( 78, 79)( 83, 84)( 85, 90)( 86, 89)( 87, 88)
( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)(105,106)
(109,136)(110,138)(111,137)(112,144)(113,143)(114,142)(115,141)(116,140)
(117,139)(118,145)(119,147)(120,146)(121,153)(122,152)(123,151)(124,150)
(125,149)(126,148)(127,154)(128,156)(129,155)(130,162)(131,161)(132,160)
(133,159)(134,158)(135,157)(163,190)(164,192)(165,191)(166,198)(167,197)
(168,196)(169,195)(170,194)(171,193)(172,199)(173,201)(174,200)(175,207)
(176,206)(177,205)(178,204)(179,203)(180,202)(181,208)(182,210)(183,209)
(184,216)(185,215)(186,214)(187,213)(188,212)(189,211)(217,325)(218,327)
(219,326)(220,333)(221,332)(222,331)(223,330)(224,329)(225,328)(226,334)
(227,336)(228,335)(229,342)(230,341)(231,340)(232,339)(233,338)(234,337)
(235,343)(236,345)(237,344)(238,351)(239,350)(240,349)(241,348)(242,347)
(243,346)(244,352)(245,354)(246,353)(247,360)(248,359)(249,358)(250,357)
(251,356)(252,355)(253,361)(254,363)(255,362)(256,369)(257,368)(258,367)
(259,366)(260,365)(261,364)(262,370)(263,372)(264,371)(265,378)(266,377)
(267,376)(268,375)(269,374)(270,373)(271,379)(272,381)(273,380)(274,387)
(275,386)(276,385)(277,384)(278,383)(279,382)(280,388)(281,390)(282,389)
(283,396)(284,395)(285,394)(286,393)(287,392)(288,391)(289,397)(290,399)
(291,398)(292,405)(293,404)(294,403)(295,402)(296,401)(297,400)(298,406)
(299,408)(300,407)(301,414)(302,413)(303,412)(304,411)(305,410)(306,409)
(307,415)(308,417)(309,416)(310,423)(311,422)(312,421)(313,420)(314,419)
(315,418)(316,424)(317,426)(318,425)(319,432)(320,431)(321,430)(322,429)
(323,428)(324,427);;
s1 := (  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,225)(  8,224)
(  9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)( 16,243)
( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)( 24,227)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)( 40,262)
( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)( 48,257)
( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,274)( 56,276)
( 57,275)( 58,271)( 59,273)( 60,272)( 61,279)( 62,278)( 63,277)( 64,292)
( 65,294)( 66,293)( 67,289)( 68,291)( 69,290)( 70,297)( 71,296)( 72,295)
( 73,283)( 74,285)( 75,284)( 76,280)( 77,282)( 78,281)( 79,288)( 80,287)
( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)( 88,306)
( 89,305)( 90,304)( 91,319)( 92,321)( 93,320)( 94,316)( 95,318)( 96,317)
( 97,324)( 98,323)( 99,322)(100,310)(101,312)(102,311)(103,307)(104,309)
(105,308)(106,315)(107,314)(108,313)(109,355)(110,357)(111,356)(112,352)
(113,354)(114,353)(115,360)(116,359)(117,358)(118,373)(119,375)(120,374)
(121,370)(122,372)(123,371)(124,378)(125,377)(126,376)(127,364)(128,366)
(129,365)(130,361)(131,363)(132,362)(133,369)(134,368)(135,367)(136,328)
(137,330)(138,329)(139,325)(140,327)(141,326)(142,333)(143,332)(144,331)
(145,346)(146,348)(147,347)(148,343)(149,345)(150,344)(151,351)(152,350)
(153,349)(154,337)(155,339)(156,338)(157,334)(158,336)(159,335)(160,342)
(161,341)(162,340)(163,409)(164,411)(165,410)(166,406)(167,408)(168,407)
(169,414)(170,413)(171,412)(172,427)(173,429)(174,428)(175,424)(176,426)
(177,425)(178,432)(179,431)(180,430)(181,418)(182,420)(183,419)(184,415)
(185,417)(186,416)(187,423)(188,422)(189,421)(190,382)(191,384)(192,383)
(193,379)(194,381)(195,380)(196,387)(197,386)(198,385)(199,400)(200,402)
(201,401)(202,397)(203,399)(204,398)(205,405)(206,404)(207,403)(208,391)
(209,393)(210,392)(211,388)(212,390)(213,389)(214,396)(215,395)(216,394);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)(112,121)
(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)(138,147)
(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)(164,173)
(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(190,199)
(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)
(217,280)(218,281)(219,282)(220,283)(221,284)(222,285)(223,286)(224,287)
(225,288)(226,271)(227,272)(228,273)(229,274)(230,275)(231,276)(232,277)
(233,278)(234,279)(235,289)(236,290)(237,291)(238,292)(239,293)(240,294)
(241,295)(242,296)(243,297)(244,307)(245,308)(246,309)(247,310)(248,311)
(249,312)(250,313)(251,314)(252,315)(253,298)(254,299)(255,300)(256,301)
(257,302)(258,303)(259,304)(260,305)(261,306)(262,316)(263,317)(264,318)
(265,319)(266,320)(267,321)(268,322)(269,323)(270,324)(325,388)(326,389)
(327,390)(328,391)(329,392)(330,393)(331,394)(332,395)(333,396)(334,379)
(335,380)(336,381)(337,382)(338,383)(339,384)(340,385)(341,386)(342,387)
(343,397)(344,398)(345,399)(346,400)(347,401)(348,402)(349,403)(350,404)
(351,405)(352,415)(353,416)(354,417)(355,418)(356,419)(357,420)(358,421)
(359,422)(360,423)(361,406)(362,407)(363,408)(364,409)(365,410)(366,411)
(367,412)(368,413)(369,414)(370,424)(371,425)(372,426)(373,427)(374,428)
(375,429)(376,430)(377,431)(378,432);;
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,
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*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*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,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)
( 15, 16)( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)
( 51, 52)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 65, 66)( 67, 72)( 68, 71)
( 69, 70)( 74, 75)( 76, 81)( 77, 80)( 78, 79)( 83, 84)( 85, 90)( 86, 89)
( 87, 88)( 92, 93)( 94, 99)( 95, 98)( 96, 97)(101,102)(103,108)(104,107)
(105,106)(109,136)(110,138)(111,137)(112,144)(113,143)(114,142)(115,141)
(116,140)(117,139)(118,145)(119,147)(120,146)(121,153)(122,152)(123,151)
(124,150)(125,149)(126,148)(127,154)(128,156)(129,155)(130,162)(131,161)
(132,160)(133,159)(134,158)(135,157)(163,190)(164,192)(165,191)(166,198)
(167,197)(168,196)(169,195)(170,194)(171,193)(172,199)(173,201)(174,200)
(175,207)(176,206)(177,205)(178,204)(179,203)(180,202)(181,208)(182,210)
(183,209)(184,216)(185,215)(186,214)(187,213)(188,212)(189,211)(217,325)
(218,327)(219,326)(220,333)(221,332)(222,331)(223,330)(224,329)(225,328)
(226,334)(227,336)(228,335)(229,342)(230,341)(231,340)(232,339)(233,338)
(234,337)(235,343)(236,345)(237,344)(238,351)(239,350)(240,349)(241,348)
(242,347)(243,346)(244,352)(245,354)(246,353)(247,360)(248,359)(249,358)
(250,357)(251,356)(252,355)(253,361)(254,363)(255,362)(256,369)(257,368)
(258,367)(259,366)(260,365)(261,364)(262,370)(263,372)(264,371)(265,378)
(266,377)(267,376)(268,375)(269,374)(270,373)(271,379)(272,381)(273,380)
(274,387)(275,386)(276,385)(277,384)(278,383)(279,382)(280,388)(281,390)
(282,389)(283,396)(284,395)(285,394)(286,393)(287,392)(288,391)(289,397)
(290,399)(291,398)(292,405)(293,404)(294,403)(295,402)(296,401)(297,400)
(298,406)(299,408)(300,407)(301,414)(302,413)(303,412)(304,411)(305,410)
(306,409)(307,415)(308,417)(309,416)(310,423)(311,422)(312,421)(313,420)
(314,419)(315,418)(316,424)(317,426)(318,425)(319,432)(320,431)(321,430)
(322,429)(323,428)(324,427);
s1 := Sym(432)!(  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,225)
(  8,224)(  9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)
( 16,243)( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)
( 24,227)( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)
( 40,262)( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)
( 48,257)( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,274)
( 56,276)( 57,275)( 58,271)( 59,273)( 60,272)( 61,279)( 62,278)( 63,277)
( 64,292)( 65,294)( 66,293)( 67,289)( 68,291)( 69,290)( 70,297)( 71,296)
( 72,295)( 73,283)( 74,285)( 75,284)( 76,280)( 77,282)( 78,281)( 79,288)
( 80,287)( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)
( 88,306)( 89,305)( 90,304)( 91,319)( 92,321)( 93,320)( 94,316)( 95,318)
( 96,317)( 97,324)( 98,323)( 99,322)(100,310)(101,312)(102,311)(103,307)
(104,309)(105,308)(106,315)(107,314)(108,313)(109,355)(110,357)(111,356)
(112,352)(113,354)(114,353)(115,360)(116,359)(117,358)(118,373)(119,375)
(120,374)(121,370)(122,372)(123,371)(124,378)(125,377)(126,376)(127,364)
(128,366)(129,365)(130,361)(131,363)(132,362)(133,369)(134,368)(135,367)
(136,328)(137,330)(138,329)(139,325)(140,327)(141,326)(142,333)(143,332)
(144,331)(145,346)(146,348)(147,347)(148,343)(149,345)(150,344)(151,351)
(152,350)(153,349)(154,337)(155,339)(156,338)(157,334)(158,336)(159,335)
(160,342)(161,341)(162,340)(163,409)(164,411)(165,410)(166,406)(167,408)
(168,407)(169,414)(170,413)(171,412)(172,427)(173,429)(174,428)(175,424)
(176,426)(177,425)(178,432)(179,431)(180,430)(181,418)(182,420)(183,419)
(184,415)(185,417)(186,416)(187,423)(188,422)(189,421)(190,382)(191,384)
(192,383)(193,379)(194,381)(195,380)(196,387)(197,386)(198,385)(199,400)
(200,402)(201,401)(202,397)(203,399)(204,398)(205,405)(206,404)(207,403)
(208,391)(209,393)(210,392)(211,388)(212,390)(213,389)(214,396)(215,395)
(216,394);
s2 := Sym(432)!(  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)
(  8, 17)(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)
( 34, 43)( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)
( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)
(112,121)(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)
(138,147)(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)
(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)
(190,199)(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)
(198,207)(217,280)(218,281)(219,282)(220,283)(221,284)(222,285)(223,286)
(224,287)(225,288)(226,271)(227,272)(228,273)(229,274)(230,275)(231,276)
(232,277)(233,278)(234,279)(235,289)(236,290)(237,291)(238,292)(239,293)
(240,294)(241,295)(242,296)(243,297)(244,307)(245,308)(246,309)(247,310)
(248,311)(249,312)(250,313)(251,314)(252,315)(253,298)(254,299)(255,300)
(256,301)(257,302)(258,303)(259,304)(260,305)(261,306)(262,316)(263,317)
(264,318)(265,319)(266,320)(267,321)(268,322)(269,323)(270,324)(325,388)
(326,389)(327,390)(328,391)(329,392)(330,393)(331,394)(332,395)(333,396)
(334,379)(335,380)(336,381)(337,382)(338,383)(339,384)(340,385)(341,386)
(342,387)(343,397)(344,398)(345,399)(346,400)(347,401)(348,402)(349,403)
(350,404)(351,405)(352,415)(353,416)(354,417)(355,418)(356,419)(357,420)
(358,421)(359,422)(360,423)(361,406)(362,407)(363,408)(364,409)(365,410)
(366,411)(367,412)(368,413)(369,414)(370,424)(371,425)(372,426)(373,427)
(374,428)(375,429)(376,430)(377,431)(378,432);
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,
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*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*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