Questions?
See the FAQ
or other info.

# Polytope of Type {144,6}

Atlas Canonical Name : {144,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,3030)
Rank : 3
Schlafli Type : {144,6}
Number of vertices, edges, etc : 144, 432, 6
Order of s0s1s2 : 144
Order of s0s1s2s1 : 6
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}*864b
3-fold quotients : {144,2}*576, {48,6}*576b
4-fold quotients : {36,6}*432b
6-fold quotients : {72,2}*288, {24,6}*288b
8-fold quotients : {18,6}*216b
9-fold quotients : {48,2}*192
12-fold quotients : {36,2}*144, {12,6}*144b
16-fold quotients : {9,6}*108
18-fold quotients : {24,2}*96
24-fold quotients : {18,2}*72, {6,6}*72c
27-fold quotients : {16,2}*64
36-fold quotients : {12,2}*48
48-fold quotients : {9,2}*36, {3,6}*36
54-fold quotients : {8,2}*32
72-fold quotients : {6,2}*24
108-fold quotients : {4,2}*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 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 19)( 11, 21)( 12, 20)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 37, 46)( 38, 48)( 39, 47)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 90)( 59, 89)( 60, 88)
( 61, 87)( 62, 86)( 63, 85)( 64,100)( 65,102)( 66,101)( 67,108)( 68,107)
( 69,106)( 70,105)( 71,104)( 72,103)( 73, 91)( 74, 93)( 75, 92)( 76, 99)
( 77, 98)( 78, 97)( 79, 96)( 80, 95)( 81, 94)(109,163)(110,165)(111,164)
(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,181)(119,183)
(120,182)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(127,172)
(128,174)(129,173)(130,180)(131,179)(132,178)(133,177)(134,176)(135,175)
(136,190)(137,192)(138,191)(139,198)(140,197)(141,196)(142,195)(143,194)
(144,193)(145,208)(146,210)(147,209)(148,216)(149,215)(150,214)(151,213)
(152,212)(153,211)(154,199)(155,201)(156,200)(157,207)(158,206)(159,205)
(160,204)(161,203)(162,202)(217,325)(218,327)(219,326)(220,333)(221,332)
(222,331)(223,330)(224,329)(225,328)(226,343)(227,345)(228,344)(229,351)
(230,350)(231,349)(232,348)(233,347)(234,346)(235,334)(236,336)(237,335)
(238,342)(239,341)(240,340)(241,339)(242,338)(243,337)(244,352)(245,354)
(246,353)(247,360)(248,359)(249,358)(250,357)(251,356)(252,355)(253,370)
(254,372)(255,371)(256,378)(257,377)(258,376)(259,375)(260,374)(261,373)
(262,361)(263,363)(264,362)(265,369)(266,368)(267,367)(268,366)(269,365)
(270,364)(271,406)(272,408)(273,407)(274,414)(275,413)(276,412)(277,411)
(278,410)(279,409)(280,424)(281,426)(282,425)(283,432)(284,431)(285,430)
(286,429)(287,428)(288,427)(289,415)(290,417)(291,416)(292,423)(293,422)
(294,421)(295,420)(296,419)(297,418)(298,379)(299,381)(300,380)(301,387)
(302,386)(303,385)(304,384)(305,383)(306,382)(307,397)(308,399)(309,398)
(310,405)(311,404)(312,403)(313,402)(314,401)(315,400)(316,388)(317,390)
(318,389)(319,396)(320,395)(321,394)(322,393)(323,392)(324,391);;
s1 := (  1,229)(  2,231)(  3,230)(  4,226)(  5,228)(  6,227)(  7,234)(  8,233)
(  9,232)( 10,220)( 11,222)( 12,221)( 13,217)( 14,219)( 15,218)( 16,225)
( 17,224)( 18,223)( 19,238)( 20,240)( 21,239)( 22,235)( 23,237)( 24,236)
( 25,243)( 26,242)( 27,241)( 28,256)( 29,258)( 30,257)( 31,253)( 32,255)
( 33,254)( 34,261)( 35,260)( 36,259)( 37,247)( 38,249)( 39,248)( 40,244)
( 41,246)( 42,245)( 43,252)( 44,251)( 45,250)( 46,265)( 47,267)( 48,266)
( 49,262)( 50,264)( 51,263)( 52,270)( 53,269)( 54,268)( 55,310)( 56,312)
( 57,311)( 58,307)( 59,309)( 60,308)( 61,315)( 62,314)( 63,313)( 64,301)
( 65,303)( 66,302)( 67,298)( 68,300)( 69,299)( 70,306)( 71,305)( 72,304)
( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,324)( 80,323)
( 81,322)( 82,283)( 83,285)( 84,284)( 85,280)( 86,282)( 87,281)( 88,288)
( 89,287)( 90,286)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)( 96,272)
( 97,279)( 98,278)( 99,277)(100,292)(101,294)(102,293)(103,289)(104,291)
(105,290)(106,297)(107,296)(108,295)(109,391)(110,393)(111,392)(112,388)
(113,390)(114,389)(115,396)(116,395)(117,394)(118,382)(119,384)(120,383)
(121,379)(122,381)(123,380)(124,387)(125,386)(126,385)(127,400)(128,402)
(129,401)(130,397)(131,399)(132,398)(133,405)(134,404)(135,403)(136,418)
(137,420)(138,419)(139,415)(140,417)(141,416)(142,423)(143,422)(144,421)
(145,409)(146,411)(147,410)(148,406)(149,408)(150,407)(151,414)(152,413)
(153,412)(154,427)(155,429)(156,428)(157,424)(158,426)(159,425)(160,432)
(161,431)(162,430)(163,337)(164,339)(165,338)(166,334)(167,336)(168,335)
(169,342)(170,341)(171,340)(172,328)(173,330)(174,329)(175,325)(176,327)
(177,326)(178,333)(179,332)(180,331)(181,346)(182,348)(183,347)(184,343)
(185,345)(186,344)(187,351)(188,350)(189,349)(190,364)(191,366)(192,365)
(193,361)(194,363)(195,362)(196,369)(197,368)(198,367)(199,355)(200,357)
(201,356)(202,352)(203,354)(204,353)(205,360)(206,359)(207,358)(208,373)
(209,375)(210,374)(211,370)(212,372)(213,371)(214,378)(215,377)(216,376);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)(173,182)
(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(199,208)
(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216)
(226,235)(227,236)(228,237)(229,238)(230,239)(231,240)(232,241)(233,242)
(234,243)(253,262)(254,263)(255,264)(256,265)(257,266)(258,267)(259,268)
(260,269)(261,270)(280,289)(281,290)(282,291)(283,292)(284,293)(285,294)
(286,295)(287,296)(288,297)(307,316)(308,317)(309,318)(310,319)(311,320)
(312,321)(313,322)(314,323)(315,324)(334,343)(335,344)(336,345)(337,346)
(338,347)(339,348)(340,349)(341,350)(342,351)(361,370)(362,371)(363,372)
(364,373)(365,374)(366,375)(367,376)(368,377)(369,378)(388,397)(389,398)
(390,399)(391,400)(392,401)(393,402)(394,403)(395,404)(396,405)(415,424)
(416,425)(417,426)(418,427)(419,428)(420,429)(421,430)(422,431)(423,432);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 10, 19)( 11, 21)( 12, 20)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 36)
( 32, 35)( 33, 34)( 37, 46)( 38, 48)( 39, 47)( 40, 54)( 41, 53)( 42, 52)
( 43, 51)( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 90)( 59, 89)
( 60, 88)( 61, 87)( 62, 86)( 63, 85)( 64,100)( 65,102)( 66,101)( 67,108)
( 68,107)( 69,106)( 70,105)( 71,104)( 72,103)( 73, 91)( 74, 93)( 75, 92)
( 76, 99)( 77, 98)( 78, 97)( 79, 96)( 80, 95)( 81, 94)(109,163)(110,165)
(111,164)(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,181)
(119,183)(120,182)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)
(127,172)(128,174)(129,173)(130,180)(131,179)(132,178)(133,177)(134,176)
(135,175)(136,190)(137,192)(138,191)(139,198)(140,197)(141,196)(142,195)
(143,194)(144,193)(145,208)(146,210)(147,209)(148,216)(149,215)(150,214)
(151,213)(152,212)(153,211)(154,199)(155,201)(156,200)(157,207)(158,206)
(159,205)(160,204)(161,203)(162,202)(217,325)(218,327)(219,326)(220,333)
(221,332)(222,331)(223,330)(224,329)(225,328)(226,343)(227,345)(228,344)
(229,351)(230,350)(231,349)(232,348)(233,347)(234,346)(235,334)(236,336)
(237,335)(238,342)(239,341)(240,340)(241,339)(242,338)(243,337)(244,352)
(245,354)(246,353)(247,360)(248,359)(249,358)(250,357)(251,356)(252,355)
(253,370)(254,372)(255,371)(256,378)(257,377)(258,376)(259,375)(260,374)
(261,373)(262,361)(263,363)(264,362)(265,369)(266,368)(267,367)(268,366)
(269,365)(270,364)(271,406)(272,408)(273,407)(274,414)(275,413)(276,412)
(277,411)(278,410)(279,409)(280,424)(281,426)(282,425)(283,432)(284,431)
(285,430)(286,429)(287,428)(288,427)(289,415)(290,417)(291,416)(292,423)
(293,422)(294,421)(295,420)(296,419)(297,418)(298,379)(299,381)(300,380)
(301,387)(302,386)(303,385)(304,384)(305,383)(306,382)(307,397)(308,399)
(309,398)(310,405)(311,404)(312,403)(313,402)(314,401)(315,400)(316,388)
(317,390)(318,389)(319,396)(320,395)(321,394)(322,393)(323,392)(324,391);
s1 := Sym(432)!(  1,229)(  2,231)(  3,230)(  4,226)(  5,228)(  6,227)(  7,234)
(  8,233)(  9,232)( 10,220)( 11,222)( 12,221)( 13,217)( 14,219)( 15,218)
( 16,225)( 17,224)( 18,223)( 19,238)( 20,240)( 21,239)( 22,235)( 23,237)
( 24,236)( 25,243)( 26,242)( 27,241)( 28,256)( 29,258)( 30,257)( 31,253)
( 32,255)( 33,254)( 34,261)( 35,260)( 36,259)( 37,247)( 38,249)( 39,248)
( 40,244)( 41,246)( 42,245)( 43,252)( 44,251)( 45,250)( 46,265)( 47,267)
( 48,266)( 49,262)( 50,264)( 51,263)( 52,270)( 53,269)( 54,268)( 55,310)
( 56,312)( 57,311)( 58,307)( 59,309)( 60,308)( 61,315)( 62,314)( 63,313)
( 64,301)( 65,303)( 66,302)( 67,298)( 68,300)( 69,299)( 70,306)( 71,305)
( 72,304)( 73,319)( 74,321)( 75,320)( 76,316)( 77,318)( 78,317)( 79,324)
( 80,323)( 81,322)( 82,283)( 83,285)( 84,284)( 85,280)( 86,282)( 87,281)
( 88,288)( 89,287)( 90,286)( 91,274)( 92,276)( 93,275)( 94,271)( 95,273)
( 96,272)( 97,279)( 98,278)( 99,277)(100,292)(101,294)(102,293)(103,289)
(104,291)(105,290)(106,297)(107,296)(108,295)(109,391)(110,393)(111,392)
(112,388)(113,390)(114,389)(115,396)(116,395)(117,394)(118,382)(119,384)
(120,383)(121,379)(122,381)(123,380)(124,387)(125,386)(126,385)(127,400)
(128,402)(129,401)(130,397)(131,399)(132,398)(133,405)(134,404)(135,403)
(136,418)(137,420)(138,419)(139,415)(140,417)(141,416)(142,423)(143,422)
(144,421)(145,409)(146,411)(147,410)(148,406)(149,408)(150,407)(151,414)
(152,413)(153,412)(154,427)(155,429)(156,428)(157,424)(158,426)(159,425)
(160,432)(161,431)(162,430)(163,337)(164,339)(165,338)(166,334)(167,336)
(168,335)(169,342)(170,341)(171,340)(172,328)(173,330)(174,329)(175,325)
(176,327)(177,326)(178,333)(179,332)(180,331)(181,346)(182,348)(183,347)
(184,343)(185,345)(186,344)(187,351)(188,350)(189,349)(190,364)(191,366)
(192,365)(193,361)(194,363)(195,362)(196,369)(197,368)(198,367)(199,355)
(200,357)(201,356)(202,352)(203,354)(204,353)(205,360)(206,359)(207,358)
(208,373)(209,375)(210,374)(211,370)(212,372)(213,371)(214,378)(215,377)
(216,376);
s2 := Sym(432)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)
(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)
(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)
(207,216)(226,235)(227,236)(228,237)(229,238)(230,239)(231,240)(232,241)
(233,242)(234,243)(253,262)(254,263)(255,264)(256,265)(257,266)(258,267)
(259,268)(260,269)(261,270)(280,289)(281,290)(282,291)(283,292)(284,293)
(285,294)(286,295)(287,296)(288,297)(307,316)(308,317)(309,318)(310,319)
(311,320)(312,321)(313,322)(314,323)(315,324)(334,343)(335,344)(336,345)
(337,346)(338,347)(339,348)(340,349)(341,350)(342,351)(361,370)(362,371)
(363,372)(364,373)(365,374)(366,375)(367,376)(368,377)(369,378)(388,397)
(389,398)(390,399)(391,400)(392,401)(393,402)(394,403)(395,404)(396,405)
(415,424)(416,425)(417,426)(418,427)(419,428)(420,429)(421,430)(422,431)
(423,432);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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