# Polytope of Type {6,6,6,4}

Atlas Canonical Name : {6,6,6,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30804)
Rank : 5
Schlafli Type : {6,6,6,4}
Number of vertices, edges, etc : 6, 18, 18, 12, 4
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
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 : {3,6,6,4}*864a, {6,6,6,2}*864a
3-fold quotients : {6,2,6,4}*576a
4-fold quotients : {3,6,6,2}*432a, {6,6,3,2}*432a
6-fold quotients : {3,2,6,4}*288a, {6,2,6,2}*288
8-fold quotients : {3,6,3,2}*216
9-fold quotients : {2,2,6,4}*192a, {6,2,2,4}*192
12-fold quotients : {3,2,6,2}*144, {6,2,3,2}*144
18-fold quotients : {3,2,2,4}*96, {2,2,6,2}*96, {6,2,2,2}*96
24-fold quotients : {3,2,3,2}*72
27-fold quotients : {2,2,2,4}*64
36-fold quotients : {2,2,3,2}*48, {3,2,2,2}*48
54-fold quotients : {2,2,2,2}*32
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)
(218,219)(221,222)(224,225)(226,235)(227,237)(228,236)(229,238)(230,240)
(231,239)(232,241)(233,243)(234,242)(245,246)(248,249)(251,252)(253,262)
(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)(261,269)
(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)(284,294)
(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)(307,316)
(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)(315,323)
(326,327)(329,330)(332,333)(334,343)(335,345)(336,344)(337,346)(338,348)
(339,347)(340,349)(341,351)(342,350)(353,354)(356,357)(359,360)(361,370)
(362,372)(363,371)(364,373)(365,375)(366,374)(367,376)(368,378)(369,377)
(380,381)(383,384)(386,387)(388,397)(389,399)(390,398)(391,400)(392,402)
(393,401)(394,403)(395,405)(396,404)(407,408)(410,411)(413,414)(415,424)
(416,426)(417,425)(418,427)(419,429)(420,428)(421,430)(422,432)(423,431);;
s1 := (  1,226)(  2,228)(  3,227)(  4,230)(  5,229)(  6,231)(  7,234)(  8,233)
(  9,232)( 10,217)( 11,219)( 12,218)( 13,221)( 14,220)( 15,222)( 16,225)
( 17,224)( 18,223)( 19,235)( 20,237)( 21,236)( 22,239)( 23,238)( 24,240)
( 25,243)( 26,242)( 27,241)( 28,253)( 29,255)( 30,254)( 31,257)( 32,256)
( 33,258)( 34,261)( 35,260)( 36,259)( 37,244)( 38,246)( 39,245)( 40,248)
( 41,247)( 42,249)( 43,252)( 44,251)( 45,250)( 46,262)( 47,264)( 48,263)
( 49,266)( 50,265)( 51,267)( 52,270)( 53,269)( 54,268)( 55,280)( 56,282)
( 57,281)( 58,284)( 59,283)( 60,285)( 61,288)( 62,287)( 63,286)( 64,271)
( 65,273)( 66,272)( 67,275)( 68,274)( 69,276)( 70,279)( 71,278)( 72,277)
( 73,289)( 74,291)( 75,290)( 76,293)( 77,292)( 78,294)( 79,297)( 80,296)
( 81,295)( 82,307)( 83,309)( 84,308)( 85,311)( 86,310)( 87,312)( 88,315)
( 89,314)( 90,313)( 91,298)( 92,300)( 93,299)( 94,302)( 95,301)( 96,303)
( 97,306)( 98,305)( 99,304)(100,316)(101,318)(102,317)(103,320)(104,319)
(105,321)(106,324)(107,323)(108,322)(109,334)(110,336)(111,335)(112,338)
(113,337)(114,339)(115,342)(116,341)(117,340)(118,325)(119,327)(120,326)
(121,329)(122,328)(123,330)(124,333)(125,332)(126,331)(127,343)(128,345)
(129,344)(130,347)(131,346)(132,348)(133,351)(134,350)(135,349)(136,361)
(137,363)(138,362)(139,365)(140,364)(141,366)(142,369)(143,368)(144,367)
(145,352)(146,354)(147,353)(148,356)(149,355)(150,357)(151,360)(152,359)
(153,358)(154,370)(155,372)(156,371)(157,374)(158,373)(159,375)(160,378)
(161,377)(162,376)(163,388)(164,390)(165,389)(166,392)(167,391)(168,393)
(169,396)(170,395)(171,394)(172,379)(173,381)(174,380)(175,383)(176,382)
(177,384)(178,387)(179,386)(180,385)(181,397)(182,399)(183,398)(184,401)
(185,400)(186,402)(187,405)(188,404)(189,403)(190,415)(191,417)(192,416)
(193,419)(194,418)(195,420)(196,423)(197,422)(198,421)(199,406)(200,408)
(201,407)(202,410)(203,409)(204,411)(205,414)(206,413)(207,412)(208,424)
(209,426)(210,425)(211,428)(212,427)(213,429)(214,432)(215,431)(216,430);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)
( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108)
(109,112)(110,114)(111,113)(116,117)(118,121)(119,123)(120,122)(125,126)
(127,130)(128,132)(129,131)(134,135)(136,139)(137,141)(138,140)(143,144)
(145,148)(146,150)(147,149)(152,153)(154,157)(155,159)(156,158)(161,162)
(163,166)(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)(179,180)
(181,184)(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)(197,198)
(199,202)(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)(215,216)
(217,220)(218,222)(219,221)(224,225)(226,229)(227,231)(228,230)(233,234)
(235,238)(236,240)(237,239)(242,243)(244,247)(245,249)(246,248)(251,252)
(253,256)(254,258)(255,257)(260,261)(262,265)(263,267)(264,266)(269,270)
(271,274)(272,276)(273,275)(278,279)(280,283)(281,285)(282,284)(287,288)
(289,292)(290,294)(291,293)(296,297)(298,301)(299,303)(300,302)(305,306)
(307,310)(308,312)(309,311)(314,315)(316,319)(317,321)(318,320)(323,324)
(325,328)(326,330)(327,329)(332,333)(334,337)(335,339)(336,338)(341,342)
(343,346)(344,348)(345,347)(350,351)(352,355)(353,357)(354,356)(359,360)
(361,364)(362,366)(363,365)(368,369)(370,373)(371,375)(372,374)(377,378)
(379,382)(380,384)(381,383)(386,387)(388,391)(389,393)(390,392)(395,396)
(397,400)(398,402)(399,401)(404,405)(406,409)(407,411)(408,410)(413,414)
(415,418)(416,420)(417,419)(422,423)(424,427)(425,429)(426,428)(431,432);;
s3 := (  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)(217,325)(218,327)(219,326)(220,331)
(221,333)(222,332)(223,328)(224,330)(225,329)(226,334)(227,336)(228,335)
(229,340)(230,342)(231,341)(232,337)(233,339)(234,338)(235,343)(236,345)
(237,344)(238,349)(239,351)(240,350)(241,346)(242,348)(243,347)(244,352)
(245,354)(246,353)(247,358)(248,360)(249,359)(250,355)(251,357)(252,356)
(253,361)(254,363)(255,362)(256,367)(257,369)(258,368)(259,364)(260,366)
(261,365)(262,370)(263,372)(264,371)(265,376)(266,378)(267,377)(268,373)
(269,375)(270,374)(271,406)(272,408)(273,407)(274,412)(275,414)(276,413)
(277,409)(278,411)(279,410)(280,415)(281,417)(282,416)(283,421)(284,423)
(285,422)(286,418)(287,420)(288,419)(289,424)(290,426)(291,425)(292,430)
(293,432)(294,431)(295,427)(296,429)(297,428)(298,379)(299,381)(300,380)
(301,385)(302,387)(303,386)(304,382)(305,384)(306,383)(307,388)(308,390)
(309,389)(310,394)(311,396)(312,395)(313,391)(314,393)(315,392)(316,397)
(317,399)(318,398)(319,403)(320,405)(321,404)(322,400)(323,402)(324,401);;
s4 := (  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)
( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)
( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)
(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)
(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)
(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)
(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)
(159,213)(160,214)(161,215)(162,216)(217,271)(218,272)(219,273)(220,274)
(221,275)(222,276)(223,277)(224,278)(225,279)(226,280)(227,281)(228,282)
(229,283)(230,284)(231,285)(232,286)(233,287)(234,288)(235,289)(236,290)
(237,291)(238,292)(239,293)(240,294)(241,295)(242,296)(243,297)(244,298)
(245,299)(246,300)(247,301)(248,302)(249,303)(250,304)(251,305)(252,306)
(253,307)(254,308)(255,309)(256,310)(257,311)(258,312)(259,313)(260,314)
(261,315)(262,316)(263,317)(264,318)(265,319)(266,320)(267,321)(268,322)
(269,323)(270,324)(325,379)(326,380)(327,381)(328,382)(329,383)(330,384)
(331,385)(332,386)(333,387)(334,388)(335,389)(336,390)(337,391)(338,392)
(339,393)(340,394)(341,395)(342,396)(343,397)(344,398)(345,399)(346,400)
(347,401)(348,402)(349,403)(350,404)(351,405)(352,406)(353,407)(354,408)
(355,409)(356,410)(357,411)(358,412)(359,413)(360,414)(361,415)(362,416)
(363,417)(364,418)(365,419)(366,420)(367,421)(368,422)(369,423)(370,424)
(371,425)(372,426)(373,427)(374,428)(375,429)(376,430)(377,431)(378,432);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)(218,219)(221,222)(224,225)(226,235)(227,237)(228,236)(229,238)
(230,240)(231,239)(232,241)(233,243)(234,242)(245,246)(248,249)(251,252)
(253,262)(254,264)(255,263)(256,265)(257,267)(258,266)(259,268)(260,270)
(261,269)(272,273)(275,276)(278,279)(280,289)(281,291)(282,290)(283,292)
(284,294)(285,293)(286,295)(287,297)(288,296)(299,300)(302,303)(305,306)
(307,316)(308,318)(309,317)(310,319)(311,321)(312,320)(313,322)(314,324)
(315,323)(326,327)(329,330)(332,333)(334,343)(335,345)(336,344)(337,346)
(338,348)(339,347)(340,349)(341,351)(342,350)(353,354)(356,357)(359,360)
(361,370)(362,372)(363,371)(364,373)(365,375)(366,374)(367,376)(368,378)
(369,377)(380,381)(383,384)(386,387)(388,397)(389,399)(390,398)(391,400)
(392,402)(393,401)(394,403)(395,405)(396,404)(407,408)(410,411)(413,414)
(415,424)(416,426)(417,425)(418,427)(419,429)(420,428)(421,430)(422,432)
(423,431);
s1 := Sym(432)!(  1,226)(  2,228)(  3,227)(  4,230)(  5,229)(  6,231)(  7,234)
(  8,233)(  9,232)( 10,217)( 11,219)( 12,218)( 13,221)( 14,220)( 15,222)
( 16,225)( 17,224)( 18,223)( 19,235)( 20,237)( 21,236)( 22,239)( 23,238)
( 24,240)( 25,243)( 26,242)( 27,241)( 28,253)( 29,255)( 30,254)( 31,257)
( 32,256)( 33,258)( 34,261)( 35,260)( 36,259)( 37,244)( 38,246)( 39,245)
( 40,248)( 41,247)( 42,249)( 43,252)( 44,251)( 45,250)( 46,262)( 47,264)
( 48,263)( 49,266)( 50,265)( 51,267)( 52,270)( 53,269)( 54,268)( 55,280)
( 56,282)( 57,281)( 58,284)( 59,283)( 60,285)( 61,288)( 62,287)( 63,286)
( 64,271)( 65,273)( 66,272)( 67,275)( 68,274)( 69,276)( 70,279)( 71,278)
( 72,277)( 73,289)( 74,291)( 75,290)( 76,293)( 77,292)( 78,294)( 79,297)
( 80,296)( 81,295)( 82,307)( 83,309)( 84,308)( 85,311)( 86,310)( 87,312)
( 88,315)( 89,314)( 90,313)( 91,298)( 92,300)( 93,299)( 94,302)( 95,301)
( 96,303)( 97,306)( 98,305)( 99,304)(100,316)(101,318)(102,317)(103,320)
(104,319)(105,321)(106,324)(107,323)(108,322)(109,334)(110,336)(111,335)
(112,338)(113,337)(114,339)(115,342)(116,341)(117,340)(118,325)(119,327)
(120,326)(121,329)(122,328)(123,330)(124,333)(125,332)(126,331)(127,343)
(128,345)(129,344)(130,347)(131,346)(132,348)(133,351)(134,350)(135,349)
(136,361)(137,363)(138,362)(139,365)(140,364)(141,366)(142,369)(143,368)
(144,367)(145,352)(146,354)(147,353)(148,356)(149,355)(150,357)(151,360)
(152,359)(153,358)(154,370)(155,372)(156,371)(157,374)(158,373)(159,375)
(160,378)(161,377)(162,376)(163,388)(164,390)(165,389)(166,392)(167,391)
(168,393)(169,396)(170,395)(171,394)(172,379)(173,381)(174,380)(175,383)
(176,382)(177,384)(178,387)(179,386)(180,385)(181,397)(182,399)(183,398)
(184,401)(185,400)(186,402)(187,405)(188,404)(189,403)(190,415)(191,417)
(192,416)(193,419)(194,418)(195,420)(196,423)(197,422)(198,421)(199,406)
(200,408)(201,407)(202,410)(203,409)(204,411)(205,414)(206,413)(207,412)
(208,424)(209,426)(210,425)(211,428)(212,427)(213,429)(214,432)(215,431)
(216,430);
s2 := Sym(432)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)
( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)
( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)
( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)
(107,108)(109,112)(110,114)(111,113)(116,117)(118,121)(119,123)(120,122)
(125,126)(127,130)(128,132)(129,131)(134,135)(136,139)(137,141)(138,140)
(143,144)(145,148)(146,150)(147,149)(152,153)(154,157)(155,159)(156,158)
(161,162)(163,166)(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)
(179,180)(181,184)(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)
(197,198)(199,202)(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)
(215,216)(217,220)(218,222)(219,221)(224,225)(226,229)(227,231)(228,230)
(233,234)(235,238)(236,240)(237,239)(242,243)(244,247)(245,249)(246,248)
(251,252)(253,256)(254,258)(255,257)(260,261)(262,265)(263,267)(264,266)
(269,270)(271,274)(272,276)(273,275)(278,279)(280,283)(281,285)(282,284)
(287,288)(289,292)(290,294)(291,293)(296,297)(298,301)(299,303)(300,302)
(305,306)(307,310)(308,312)(309,311)(314,315)(316,319)(317,321)(318,320)
(323,324)(325,328)(326,330)(327,329)(332,333)(334,337)(335,339)(336,338)
(341,342)(343,346)(344,348)(345,347)(350,351)(352,355)(353,357)(354,356)
(359,360)(361,364)(362,366)(363,365)(368,369)(370,373)(371,375)(372,374)
(377,378)(379,382)(380,384)(381,383)(386,387)(388,391)(389,393)(390,392)
(395,396)(397,400)(398,402)(399,401)(404,405)(406,409)(407,411)(408,410)
(413,414)(415,418)(416,420)(417,419)(422,423)(424,427)(425,429)(426,428)
(431,432);
s3 := 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)(217,325)(218,327)(219,326)
(220,331)(221,333)(222,332)(223,328)(224,330)(225,329)(226,334)(227,336)
(228,335)(229,340)(230,342)(231,341)(232,337)(233,339)(234,338)(235,343)
(236,345)(237,344)(238,349)(239,351)(240,350)(241,346)(242,348)(243,347)
(244,352)(245,354)(246,353)(247,358)(248,360)(249,359)(250,355)(251,357)
(252,356)(253,361)(254,363)(255,362)(256,367)(257,369)(258,368)(259,364)
(260,366)(261,365)(262,370)(263,372)(264,371)(265,376)(266,378)(267,377)
(268,373)(269,375)(270,374)(271,406)(272,408)(273,407)(274,412)(275,414)
(276,413)(277,409)(278,411)(279,410)(280,415)(281,417)(282,416)(283,421)
(284,423)(285,422)(286,418)(287,420)(288,419)(289,424)(290,426)(291,425)
(292,430)(293,432)(294,431)(295,427)(296,429)(297,428)(298,379)(299,381)
(300,380)(301,385)(302,387)(303,386)(304,382)(305,384)(306,383)(307,388)
(308,390)(309,389)(310,394)(311,396)(312,395)(313,391)(314,393)(315,392)
(316,397)(317,399)(318,398)(319,403)(320,405)(321,404)(322,400)(323,402)
(324,401);
s4 := Sym(432)!(  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)
(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)
( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)
( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)
( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)
(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)
(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)
(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)
(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)
(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)
(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)
(158,212)(159,213)(160,214)(161,215)(162,216)(217,271)(218,272)(219,273)
(220,274)(221,275)(222,276)(223,277)(224,278)(225,279)(226,280)(227,281)
(228,282)(229,283)(230,284)(231,285)(232,286)(233,287)(234,288)(235,289)
(236,290)(237,291)(238,292)(239,293)(240,294)(241,295)(242,296)(243,297)
(244,298)(245,299)(246,300)(247,301)(248,302)(249,303)(250,304)(251,305)
(252,306)(253,307)(254,308)(255,309)(256,310)(257,311)(258,312)(259,313)
(260,314)(261,315)(262,316)(263,317)(264,318)(265,319)(266,320)(267,321)
(268,322)(269,323)(270,324)(325,379)(326,380)(327,381)(328,382)(329,383)
(330,384)(331,385)(332,386)(333,387)(334,388)(335,389)(336,390)(337,391)
(338,392)(339,393)(340,394)(341,395)(342,396)(343,397)(344,398)(345,399)
(346,400)(347,401)(348,402)(349,403)(350,404)(351,405)(352,406)(353,407)
(354,408)(355,409)(356,410)(357,411)(358,412)(359,413)(360,414)(361,415)
(362,416)(363,417)(364,418)(365,419)(366,420)(367,421)(368,422)(369,423)
(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,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope