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# Polytope of Type {108,4}

Atlas Canonical Name : {108,4}*864b
if this polytope has a name.
Group : SmallGroup(864,613)
Rank : 3
Schlafli Type : {108,4}
Number of vertices, edges, etc : 108, 216, 4
Order of s0s1s2 : 108
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{108,4,2} of size 1728
Vertex Figure Of :
{2,108,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {54,4}*432b
3-fold quotients : {36,4}*288b
4-fold quotients : {27,4}*216
6-fold quotients : {18,4}*144b
9-fold quotients : {12,4}*96b
12-fold quotients : {9,4}*72
18-fold quotients : {6,4}*48c
36-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {216,4}*1728c, {216,4}*1728d, {108,4}*1728b
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 33)( 14, 35)( 15, 34)
( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)( 23, 26)
( 24, 28)( 37,105)( 38,107)( 39,106)( 40,108)( 41,101)( 42,103)( 43,102)
( 44,104)( 45, 97)( 46, 99)( 47, 98)( 48,100)( 49, 93)( 50, 95)( 51, 94)
( 52, 96)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)( 59, 86)
( 60, 88)( 61, 81)( 62, 83)( 63, 82)( 64, 84)( 65, 77)( 66, 79)( 67, 78)
( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)
(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(145,213)(146,215)
(147,214)(148,216)(149,209)(150,211)(151,210)(152,212)(153,205)(154,207)
(155,206)(156,208)(157,201)(158,203)(159,202)(160,204)(161,197)(162,199)
(163,198)(164,200)(165,193)(166,195)(167,194)(168,196)(169,189)(170,191)
(171,190)(172,192)(173,185)(174,187)(175,186)(176,188)(177,181)(178,183)
(179,182)(180,184)(217,325)(218,327)(219,326)(220,328)(221,333)(222,335)
(223,334)(224,336)(225,329)(226,331)(227,330)(228,332)(229,357)(230,359)
(231,358)(232,360)(233,353)(234,355)(235,354)(236,356)(237,349)(238,351)
(239,350)(240,352)(241,345)(242,347)(243,346)(244,348)(245,341)(246,343)
(247,342)(248,344)(249,337)(250,339)(251,338)(252,340)(253,429)(254,431)
(255,430)(256,432)(257,425)(258,427)(259,426)(260,428)(261,421)(262,423)
(263,422)(264,424)(265,417)(266,419)(267,418)(268,420)(269,413)(270,415)
(271,414)(272,416)(273,409)(274,411)(275,410)(276,412)(277,405)(278,407)
(279,406)(280,408)(281,401)(282,403)(283,402)(284,404)(285,397)(286,399)
(287,398)(288,400)(289,393)(290,395)(291,394)(292,396)(293,389)(294,391)
(295,390)(296,392)(297,385)(298,387)(299,386)(300,388)(301,381)(302,383)
(303,382)(304,384)(305,377)(306,379)(307,378)(308,380)(309,373)(310,375)
(311,374)(312,376)(313,369)(314,371)(315,370)(316,372)(317,365)(318,367)
(319,366)(320,368)(321,361)(322,363)(323,362)(324,364);;
s1 := (  1,253)(  2,254)(  3,256)(  4,255)(  5,261)(  6,262)(  7,264)(  8,263)
(  9,257)( 10,258)( 11,260)( 12,259)( 13,285)( 14,286)( 15,288)( 16,287)
( 17,281)( 18,282)( 19,284)( 20,283)( 21,277)( 22,278)( 23,280)( 24,279)
( 25,273)( 26,274)( 27,276)( 28,275)( 29,269)( 30,270)( 31,272)( 32,271)
( 33,265)( 34,266)( 35,268)( 36,267)( 37,217)( 38,218)( 39,220)( 40,219)
( 41,225)( 42,226)( 43,228)( 44,227)( 45,221)( 46,222)( 47,224)( 48,223)
( 49,249)( 50,250)( 51,252)( 52,251)( 53,245)( 54,246)( 55,248)( 56,247)
( 57,241)( 58,242)( 59,244)( 60,243)( 61,237)( 62,238)( 63,240)( 64,239)
( 65,233)( 66,234)( 67,236)( 68,235)( 69,229)( 70,230)( 71,232)( 72,231)
( 73,321)( 74,322)( 75,324)( 76,323)( 77,317)( 78,318)( 79,320)( 80,319)
( 81,313)( 82,314)( 83,316)( 84,315)( 85,309)( 86,310)( 87,312)( 88,311)
( 89,305)( 90,306)( 91,308)( 92,307)( 93,301)( 94,302)( 95,304)( 96,303)
( 97,297)( 98,298)( 99,300)(100,299)(101,293)(102,294)(103,296)(104,295)
(105,289)(106,290)(107,292)(108,291)(109,361)(110,362)(111,364)(112,363)
(113,369)(114,370)(115,372)(116,371)(117,365)(118,366)(119,368)(120,367)
(121,393)(122,394)(123,396)(124,395)(125,389)(126,390)(127,392)(128,391)
(129,385)(130,386)(131,388)(132,387)(133,381)(134,382)(135,384)(136,383)
(137,377)(138,378)(139,380)(140,379)(141,373)(142,374)(143,376)(144,375)
(145,325)(146,326)(147,328)(148,327)(149,333)(150,334)(151,336)(152,335)
(153,329)(154,330)(155,332)(156,331)(157,357)(158,358)(159,360)(160,359)
(161,353)(162,354)(163,356)(164,355)(165,349)(166,350)(167,352)(168,351)
(169,345)(170,346)(171,348)(172,347)(173,341)(174,342)(175,344)(176,343)
(177,337)(178,338)(179,340)(180,339)(181,429)(182,430)(183,432)(184,431)
(185,425)(186,426)(187,428)(188,427)(189,421)(190,422)(191,424)(192,423)
(193,417)(194,418)(195,420)(196,419)(197,413)(198,414)(199,416)(200,415)
(201,409)(202,410)(203,412)(204,411)(205,405)(206,406)(207,408)(208,407)
(209,401)(210,402)(211,404)(212,403)(213,397)(214,398)(215,400)(216,399);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239)
(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)(254,255)
(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)(270,271)
(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)(286,287)
(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)(302,303)
(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)(318,319)
(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)(334,335)
(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)(350,351)
(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)(366,367)
(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)(382,383)
(385,388)(386,387)(389,392)(390,391)(393,396)(394,395)(397,400)(398,399)
(401,404)(402,403)(405,408)(406,407)(409,412)(410,411)(413,416)(414,415)
(417,420)(418,419)(421,424)(422,423)(425,428)(426,427)(429,432)(430,431);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s1*s0*s2*s1*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*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*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)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 33)( 14, 35)
( 15, 34)( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)
( 23, 26)( 24, 28)( 37,105)( 38,107)( 39,106)( 40,108)( 41,101)( 42,103)
( 43,102)( 44,104)( 45, 97)( 46, 99)( 47, 98)( 48,100)( 49, 93)( 50, 95)
( 51, 94)( 52, 96)( 53, 89)( 54, 91)( 55, 90)( 56, 92)( 57, 85)( 58, 87)
( 59, 86)( 60, 88)( 61, 81)( 62, 83)( 63, 82)( 64, 84)( 65, 77)( 66, 79)
( 67, 78)( 68, 80)( 69, 73)( 70, 75)( 71, 74)( 72, 76)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)
(126,139)(127,138)(128,140)(129,133)(130,135)(131,134)(132,136)(145,213)
(146,215)(147,214)(148,216)(149,209)(150,211)(151,210)(152,212)(153,205)
(154,207)(155,206)(156,208)(157,201)(158,203)(159,202)(160,204)(161,197)
(162,199)(163,198)(164,200)(165,193)(166,195)(167,194)(168,196)(169,189)
(170,191)(171,190)(172,192)(173,185)(174,187)(175,186)(176,188)(177,181)
(178,183)(179,182)(180,184)(217,325)(218,327)(219,326)(220,328)(221,333)
(222,335)(223,334)(224,336)(225,329)(226,331)(227,330)(228,332)(229,357)
(230,359)(231,358)(232,360)(233,353)(234,355)(235,354)(236,356)(237,349)
(238,351)(239,350)(240,352)(241,345)(242,347)(243,346)(244,348)(245,341)
(246,343)(247,342)(248,344)(249,337)(250,339)(251,338)(252,340)(253,429)
(254,431)(255,430)(256,432)(257,425)(258,427)(259,426)(260,428)(261,421)
(262,423)(263,422)(264,424)(265,417)(266,419)(267,418)(268,420)(269,413)
(270,415)(271,414)(272,416)(273,409)(274,411)(275,410)(276,412)(277,405)
(278,407)(279,406)(280,408)(281,401)(282,403)(283,402)(284,404)(285,397)
(286,399)(287,398)(288,400)(289,393)(290,395)(291,394)(292,396)(293,389)
(294,391)(295,390)(296,392)(297,385)(298,387)(299,386)(300,388)(301,381)
(302,383)(303,382)(304,384)(305,377)(306,379)(307,378)(308,380)(309,373)
(310,375)(311,374)(312,376)(313,369)(314,371)(315,370)(316,372)(317,365)
(318,367)(319,366)(320,368)(321,361)(322,363)(323,362)(324,364);
s1 := Sym(432)!(  1,253)(  2,254)(  3,256)(  4,255)(  5,261)(  6,262)(  7,264)
(  8,263)(  9,257)( 10,258)( 11,260)( 12,259)( 13,285)( 14,286)( 15,288)
( 16,287)( 17,281)( 18,282)( 19,284)( 20,283)( 21,277)( 22,278)( 23,280)
( 24,279)( 25,273)( 26,274)( 27,276)( 28,275)( 29,269)( 30,270)( 31,272)
( 32,271)( 33,265)( 34,266)( 35,268)( 36,267)( 37,217)( 38,218)( 39,220)
( 40,219)( 41,225)( 42,226)( 43,228)( 44,227)( 45,221)( 46,222)( 47,224)
( 48,223)( 49,249)( 50,250)( 51,252)( 52,251)( 53,245)( 54,246)( 55,248)
( 56,247)( 57,241)( 58,242)( 59,244)( 60,243)( 61,237)( 62,238)( 63,240)
( 64,239)( 65,233)( 66,234)( 67,236)( 68,235)( 69,229)( 70,230)( 71,232)
( 72,231)( 73,321)( 74,322)( 75,324)( 76,323)( 77,317)( 78,318)( 79,320)
( 80,319)( 81,313)( 82,314)( 83,316)( 84,315)( 85,309)( 86,310)( 87,312)
( 88,311)( 89,305)( 90,306)( 91,308)( 92,307)( 93,301)( 94,302)( 95,304)
( 96,303)( 97,297)( 98,298)( 99,300)(100,299)(101,293)(102,294)(103,296)
(104,295)(105,289)(106,290)(107,292)(108,291)(109,361)(110,362)(111,364)
(112,363)(113,369)(114,370)(115,372)(116,371)(117,365)(118,366)(119,368)
(120,367)(121,393)(122,394)(123,396)(124,395)(125,389)(126,390)(127,392)
(128,391)(129,385)(130,386)(131,388)(132,387)(133,381)(134,382)(135,384)
(136,383)(137,377)(138,378)(139,380)(140,379)(141,373)(142,374)(143,376)
(144,375)(145,325)(146,326)(147,328)(148,327)(149,333)(150,334)(151,336)
(152,335)(153,329)(154,330)(155,332)(156,331)(157,357)(158,358)(159,360)
(160,359)(161,353)(162,354)(163,356)(164,355)(165,349)(166,350)(167,352)
(168,351)(169,345)(170,346)(171,348)(172,347)(173,341)(174,342)(175,344)
(176,343)(177,337)(178,338)(179,340)(180,339)(181,429)(182,430)(183,432)
(184,431)(185,425)(186,426)(187,428)(188,427)(189,421)(190,422)(191,424)
(192,423)(193,417)(194,418)(195,420)(196,419)(197,413)(198,414)(199,416)
(200,415)(201,409)(202,410)(203,412)(204,411)(205,405)(206,406)(207,408)
(208,407)(209,401)(210,402)(211,404)(212,403)(213,397)(214,398)(215,400)
(216,399);
s2 := Sym(432)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239)(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)
(254,255)(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)
(270,271)(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)
(286,287)(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)
(302,303)(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)
(318,319)(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)
(334,335)(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)
(350,351)(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)
(366,367)(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)
(382,383)(385,388)(386,387)(389,392)(390,391)(393,396)(394,395)(397,400)
(398,399)(401,404)(402,403)(405,408)(406,407)(409,412)(410,411)(413,416)
(414,415)(417,420)(418,419)(421,424)(422,423)(425,428)(426,427)(429,432)
(430,431);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s1*s0*s2*s1*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*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*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