Questions?
See the FAQ
or other info.

# Polytope of Type {4,54}

Atlas Canonical Name : {4,54}*864
if this polytope has a name.
Group : SmallGroup(864,1916)
Rank : 3
Schlafli Type : {4,54}
Number of vertices, edges, etc : 8, 216, 108
Order of s0s1s2 : 54
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,54,2} of size 1728
Vertex Figure Of :
{2,4,54} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,27}*432, {4,54}*432b, {4,54}*432c
3-fold quotients : {4,18}*288
4-fold quotients : {4,27}*216, {2,54}*216
6-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
8-fold quotients : {2,27}*108
9-fold quotients : {4,6}*96
12-fold quotients : {4,9}*72, {2,18}*72
18-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
24-fold quotients : {2,9}*36
36-fold quotients : {4,3}*24, {2,6}*24
72-fold quotients : {2,3}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c
Permutation Representation (GAP) :
```s0 := (  1,219)(  2,220)(  3,217)(  4,218)(  5,223)(  6,224)(  7,221)(  8,222)
(  9,227)( 10,228)( 11,225)( 12,226)( 13,231)( 14,232)( 15,229)( 16,230)
( 17,235)( 18,236)( 19,233)( 20,234)( 21,239)( 22,240)( 23,237)( 24,238)
( 25,243)( 26,244)( 27,241)( 28,242)( 29,247)( 30,248)( 31,245)( 32,246)
( 33,251)( 34,252)( 35,249)( 36,250)( 37,255)( 38,256)( 39,253)( 40,254)
( 41,259)( 42,260)( 43,257)( 44,258)( 45,263)( 46,264)( 47,261)( 48,262)
( 49,267)( 50,268)( 51,265)( 52,266)( 53,271)( 54,272)( 55,269)( 56,270)
( 57,275)( 58,276)( 59,273)( 60,274)( 61,279)( 62,280)( 63,277)( 64,278)
( 65,283)( 66,284)( 67,281)( 68,282)( 69,287)( 70,288)( 71,285)( 72,286)
( 73,291)( 74,292)( 75,289)( 76,290)( 77,295)( 78,296)( 79,293)( 80,294)
( 81,299)( 82,300)( 83,297)( 84,298)( 85,303)( 86,304)( 87,301)( 88,302)
( 89,307)( 90,308)( 91,305)( 92,306)( 93,311)( 94,312)( 95,309)( 96,310)
( 97,315)( 98,316)( 99,313)(100,314)(101,319)(102,320)(103,317)(104,318)
(105,323)(106,324)(107,321)(108,322)(109,327)(110,328)(111,325)(112,326)
(113,331)(114,332)(115,329)(116,330)(117,335)(118,336)(119,333)(120,334)
(121,339)(122,340)(123,337)(124,338)(125,343)(126,344)(127,341)(128,342)
(129,347)(130,348)(131,345)(132,346)(133,351)(134,352)(135,349)(136,350)
(137,355)(138,356)(139,353)(140,354)(141,359)(142,360)(143,357)(144,358)
(145,363)(146,364)(147,361)(148,362)(149,367)(150,368)(151,365)(152,366)
(153,371)(154,372)(155,369)(156,370)(157,375)(158,376)(159,373)(160,374)
(161,379)(162,380)(163,377)(164,378)(165,383)(166,384)(167,381)(168,382)
(169,387)(170,388)(171,385)(172,386)(173,391)(174,392)(175,389)(176,390)
(177,395)(178,396)(179,393)(180,394)(181,399)(182,400)(183,397)(184,398)
(185,403)(186,404)(187,401)(188,402)(189,407)(190,408)(191,405)(192,406)
(193,411)(194,412)(195,409)(196,410)(197,415)(198,416)(199,413)(200,414)
(201,419)(202,420)(203,417)(204,418)(205,423)(206,424)(207,421)(208,422)
(209,427)(210,428)(211,425)(212,426)(213,431)(214,432)(215,429)(216,430);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)( 43,106)
( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)( 51, 86)
( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)
( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)( 67, 82)
( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)
(127,134)(128,136)(129,141)(130,143)(131,142)(132,144)(145,205)(146,207)
(147,206)(148,208)(149,213)(150,215)(151,214)(152,216)(153,209)(154,211)
(155,210)(156,212)(157,193)(158,195)(159,194)(160,196)(161,201)(162,203)
(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,181)(170,183)
(171,182)(172,184)(173,189)(174,191)(175,190)(176,192)(177,185)(178,187)
(179,186)(180,188)(218,219)(221,225)(222,227)(223,226)(224,228)(229,245)
(230,247)(231,246)(232,248)(233,241)(234,243)(235,242)(236,244)(237,249)
(238,251)(239,250)(240,252)(253,313)(254,315)(255,314)(256,316)(257,321)
(258,323)(259,322)(260,324)(261,317)(262,319)(263,318)(264,320)(265,301)
(266,303)(267,302)(268,304)(269,309)(270,311)(271,310)(272,312)(273,305)
(274,307)(275,306)(276,308)(277,289)(278,291)(279,290)(280,292)(281,297)
(282,299)(283,298)(284,300)(285,293)(286,295)(287,294)(288,296)(326,327)
(329,333)(330,335)(331,334)(332,336)(337,353)(338,355)(339,354)(340,356)
(341,349)(342,351)(343,350)(344,352)(345,357)(346,359)(347,358)(348,360)
(361,421)(362,423)(363,422)(364,424)(365,429)(366,431)(367,430)(368,432)
(369,425)(370,427)(371,426)(372,428)(373,409)(374,411)(375,410)(376,412)
(377,417)(378,419)(379,418)(380,420)(381,413)(382,415)(383,414)(384,416)
(385,397)(386,399)(387,398)(388,400)(389,405)(390,407)(391,406)(392,408)
(393,401)(394,403)(395,402)(396,404);;
s2 := (  1,181)(  2,184)(  3,183)(  4,182)(  5,189)(  6,192)(  7,191)(  8,190)
(  9,185)( 10,188)( 11,187)( 12,186)( 13,209)( 14,212)( 15,211)( 16,210)
( 17,205)( 18,208)( 19,207)( 20,206)( 21,213)( 22,216)( 23,215)( 24,214)
( 25,197)( 26,200)( 27,199)( 28,198)( 29,193)( 30,196)( 31,195)( 32,194)
( 33,201)( 34,204)( 35,203)( 36,202)( 37,145)( 38,148)( 39,147)( 40,146)
( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)( 48,150)
( 49,173)( 50,176)( 51,175)( 52,174)( 53,169)( 54,172)( 55,171)( 56,170)
( 57,177)( 58,180)( 59,179)( 60,178)( 61,161)( 62,164)( 63,163)( 64,162)
( 65,157)( 66,160)( 67,159)( 68,158)( 69,165)( 70,168)( 71,167)( 72,166)
( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)( 80,118)
( 81,113)( 82,116)( 83,115)( 84,114)( 85,137)( 86,140)( 87,139)( 88,138)
( 89,133)( 90,136)( 91,135)( 92,134)( 93,141)( 94,144)( 95,143)( 96,142)
( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)(104,122)
(105,129)(106,132)(107,131)(108,130)(217,397)(218,400)(219,399)(220,398)
(221,405)(222,408)(223,407)(224,406)(225,401)(226,404)(227,403)(228,402)
(229,425)(230,428)(231,427)(232,426)(233,421)(234,424)(235,423)(236,422)
(237,429)(238,432)(239,431)(240,430)(241,413)(242,416)(243,415)(244,414)
(245,409)(246,412)(247,411)(248,410)(249,417)(250,420)(251,419)(252,418)
(253,361)(254,364)(255,363)(256,362)(257,369)(258,372)(259,371)(260,370)
(261,365)(262,368)(263,367)(264,366)(265,389)(266,392)(267,391)(268,390)
(269,385)(270,388)(271,387)(272,386)(273,393)(274,396)(275,395)(276,394)
(277,377)(278,380)(279,379)(280,378)(281,373)(282,376)(283,375)(284,374)
(285,381)(286,384)(287,383)(288,382)(289,325)(290,328)(291,327)(292,326)
(293,333)(294,336)(295,335)(296,334)(297,329)(298,332)(299,331)(300,330)
(301,353)(302,356)(303,355)(304,354)(305,349)(306,352)(307,351)(308,350)
(309,357)(310,360)(311,359)(312,358)(313,341)(314,344)(315,343)(316,342)
(317,337)(318,340)(319,339)(320,338)(321,345)(322,348)(323,347)(324,346);;
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*s2*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(  1,219)(  2,220)(  3,217)(  4,218)(  5,223)(  6,224)(  7,221)
(  8,222)(  9,227)( 10,228)( 11,225)( 12,226)( 13,231)( 14,232)( 15,229)
( 16,230)( 17,235)( 18,236)( 19,233)( 20,234)( 21,239)( 22,240)( 23,237)
( 24,238)( 25,243)( 26,244)( 27,241)( 28,242)( 29,247)( 30,248)( 31,245)
( 32,246)( 33,251)( 34,252)( 35,249)( 36,250)( 37,255)( 38,256)( 39,253)
( 40,254)( 41,259)( 42,260)( 43,257)( 44,258)( 45,263)( 46,264)( 47,261)
( 48,262)( 49,267)( 50,268)( 51,265)( 52,266)( 53,271)( 54,272)( 55,269)
( 56,270)( 57,275)( 58,276)( 59,273)( 60,274)( 61,279)( 62,280)( 63,277)
( 64,278)( 65,283)( 66,284)( 67,281)( 68,282)( 69,287)( 70,288)( 71,285)
( 72,286)( 73,291)( 74,292)( 75,289)( 76,290)( 77,295)( 78,296)( 79,293)
( 80,294)( 81,299)( 82,300)( 83,297)( 84,298)( 85,303)( 86,304)( 87,301)
( 88,302)( 89,307)( 90,308)( 91,305)( 92,306)( 93,311)( 94,312)( 95,309)
( 96,310)( 97,315)( 98,316)( 99,313)(100,314)(101,319)(102,320)(103,317)
(104,318)(105,323)(106,324)(107,321)(108,322)(109,327)(110,328)(111,325)
(112,326)(113,331)(114,332)(115,329)(116,330)(117,335)(118,336)(119,333)
(120,334)(121,339)(122,340)(123,337)(124,338)(125,343)(126,344)(127,341)
(128,342)(129,347)(130,348)(131,345)(132,346)(133,351)(134,352)(135,349)
(136,350)(137,355)(138,356)(139,353)(140,354)(141,359)(142,360)(143,357)
(144,358)(145,363)(146,364)(147,361)(148,362)(149,367)(150,368)(151,365)
(152,366)(153,371)(154,372)(155,369)(156,370)(157,375)(158,376)(159,373)
(160,374)(161,379)(162,380)(163,377)(164,378)(165,383)(166,384)(167,381)
(168,382)(169,387)(170,388)(171,385)(172,386)(173,391)(174,392)(175,389)
(176,390)(177,395)(178,396)(179,393)(180,394)(181,399)(182,400)(183,397)
(184,398)(185,403)(186,404)(187,401)(188,402)(189,407)(190,408)(191,405)
(192,406)(193,411)(194,412)(195,409)(196,410)(197,415)(198,416)(199,413)
(200,414)(201,419)(202,420)(203,417)(204,418)(205,423)(206,424)(207,421)
(208,422)(209,427)(210,428)(211,425)(212,426)(213,431)(214,432)(215,429)
(216,430);
s1 := Sym(432)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)
( 43,106)( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)
( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)
( 59, 90)( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)
( 67, 82)( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)
(126,135)(127,134)(128,136)(129,141)(130,143)(131,142)(132,144)(145,205)
(146,207)(147,206)(148,208)(149,213)(150,215)(151,214)(152,216)(153,209)
(154,211)(155,210)(156,212)(157,193)(158,195)(159,194)(160,196)(161,201)
(162,203)(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,181)
(170,183)(171,182)(172,184)(173,189)(174,191)(175,190)(176,192)(177,185)
(178,187)(179,186)(180,188)(218,219)(221,225)(222,227)(223,226)(224,228)
(229,245)(230,247)(231,246)(232,248)(233,241)(234,243)(235,242)(236,244)
(237,249)(238,251)(239,250)(240,252)(253,313)(254,315)(255,314)(256,316)
(257,321)(258,323)(259,322)(260,324)(261,317)(262,319)(263,318)(264,320)
(265,301)(266,303)(267,302)(268,304)(269,309)(270,311)(271,310)(272,312)
(273,305)(274,307)(275,306)(276,308)(277,289)(278,291)(279,290)(280,292)
(281,297)(282,299)(283,298)(284,300)(285,293)(286,295)(287,294)(288,296)
(326,327)(329,333)(330,335)(331,334)(332,336)(337,353)(338,355)(339,354)
(340,356)(341,349)(342,351)(343,350)(344,352)(345,357)(346,359)(347,358)
(348,360)(361,421)(362,423)(363,422)(364,424)(365,429)(366,431)(367,430)
(368,432)(369,425)(370,427)(371,426)(372,428)(373,409)(374,411)(375,410)
(376,412)(377,417)(378,419)(379,418)(380,420)(381,413)(382,415)(383,414)
(384,416)(385,397)(386,399)(387,398)(388,400)(389,405)(390,407)(391,406)
(392,408)(393,401)(394,403)(395,402)(396,404);
s2 := Sym(432)!(  1,181)(  2,184)(  3,183)(  4,182)(  5,189)(  6,192)(  7,191)
(  8,190)(  9,185)( 10,188)( 11,187)( 12,186)( 13,209)( 14,212)( 15,211)
( 16,210)( 17,205)( 18,208)( 19,207)( 20,206)( 21,213)( 22,216)( 23,215)
( 24,214)( 25,197)( 26,200)( 27,199)( 28,198)( 29,193)( 30,196)( 31,195)
( 32,194)( 33,201)( 34,204)( 35,203)( 36,202)( 37,145)( 38,148)( 39,147)
( 40,146)( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)
( 48,150)( 49,173)( 50,176)( 51,175)( 52,174)( 53,169)( 54,172)( 55,171)
( 56,170)( 57,177)( 58,180)( 59,179)( 60,178)( 61,161)( 62,164)( 63,163)
( 64,162)( 65,157)( 66,160)( 67,159)( 68,158)( 69,165)( 70,168)( 71,167)
( 72,166)( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)
( 80,118)( 81,113)( 82,116)( 83,115)( 84,114)( 85,137)( 86,140)( 87,139)
( 88,138)( 89,133)( 90,136)( 91,135)( 92,134)( 93,141)( 94,144)( 95,143)
( 96,142)( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)
(104,122)(105,129)(106,132)(107,131)(108,130)(217,397)(218,400)(219,399)
(220,398)(221,405)(222,408)(223,407)(224,406)(225,401)(226,404)(227,403)
(228,402)(229,425)(230,428)(231,427)(232,426)(233,421)(234,424)(235,423)
(236,422)(237,429)(238,432)(239,431)(240,430)(241,413)(242,416)(243,415)
(244,414)(245,409)(246,412)(247,411)(248,410)(249,417)(250,420)(251,419)
(252,418)(253,361)(254,364)(255,363)(256,362)(257,369)(258,372)(259,371)
(260,370)(261,365)(262,368)(263,367)(264,366)(265,389)(266,392)(267,391)
(268,390)(269,385)(270,388)(271,387)(272,386)(273,393)(274,396)(275,395)
(276,394)(277,377)(278,380)(279,379)(280,378)(281,373)(282,376)(283,375)
(284,374)(285,381)(286,384)(287,383)(288,382)(289,325)(290,328)(291,327)
(292,326)(293,333)(294,336)(295,335)(296,334)(297,329)(298,332)(299,331)
(300,330)(301,353)(302,356)(303,355)(304,354)(305,349)(306,352)(307,351)
(308,350)(309,357)(310,360)(311,359)(312,358)(313,341)(314,344)(315,343)
(316,342)(317,337)(318,340)(319,339)(320,338)(321,345)(322,348)(323,347)
(324,346);
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*s2*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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