Questions?
See the FAQ
or other info.

# Polytope of Type {72,6}

Atlas Canonical Name : {72,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30201)
Rank : 3
Schlafli Type : {72,6}
Number of vertices, edges, etc : 144, 432, 12
Order of s0s1s2 : 36
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {36,6}*864
3-fold quotients : {24,6}*576e
4-fold quotients : {36,6}*432c
6-fold quotients : {12,6}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {8,6}*192b
12-fold quotients : {12,6}*144d
18-fold quotients : {8,3}*96, {4,6}*96
24-fold quotients : {18,2}*72, {6,6}*72a
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {9,2}*36
72-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
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 := (  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)
(  9,237)( 10,238)( 11,240)( 12,239)( 13,234)( 14,233)( 15,235)( 16,236)
( 17,229)( 18,230)( 19,232)( 20,231)( 21,226)( 22,225)( 23,227)( 24,228)
( 25,285)( 26,286)( 27,288)( 28,287)( 29,282)( 30,281)( 31,283)( 32,284)
( 33,277)( 34,278)( 35,280)( 36,279)( 37,274)( 38,273)( 39,275)( 40,276)
( 41,269)( 42,270)( 43,272)( 44,271)( 45,266)( 46,265)( 47,267)( 48,268)
( 49,261)( 50,262)( 51,264)( 52,263)( 53,258)( 54,257)( 55,259)( 56,260)
( 57,253)( 58,254)( 59,256)( 60,255)( 61,250)( 62,249)( 63,251)( 64,252)
( 65,245)( 66,246)( 67,248)( 68,247)( 69,242)( 70,241)( 71,243)( 72,244)
( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)
( 81,309)( 82,310)( 83,312)( 84,311)( 85,306)( 86,305)( 87,307)( 88,308)
( 89,301)( 90,302)( 91,304)( 92,303)( 93,298)( 94,297)( 95,299)( 96,300)
( 97,357)( 98,358)( 99,360)(100,359)(101,354)(102,353)(103,355)(104,356)
(105,349)(106,350)(107,352)(108,351)(109,346)(110,345)(111,347)(112,348)
(113,341)(114,342)(115,344)(116,343)(117,338)(118,337)(119,339)(120,340)
(121,333)(122,334)(123,336)(124,335)(125,330)(126,329)(127,331)(128,332)
(129,325)(130,326)(131,328)(132,327)(133,322)(134,321)(135,323)(136,324)
(137,317)(138,318)(139,320)(140,319)(141,314)(142,313)(143,315)(144,316)
(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)
(153,381)(154,382)(155,384)(156,383)(157,378)(158,377)(159,379)(160,380)
(161,373)(162,374)(163,376)(164,375)(165,370)(166,369)(167,371)(168,372)
(169,429)(170,430)(171,432)(172,431)(173,426)(174,425)(175,427)(176,428)
(177,421)(178,422)(179,424)(180,423)(181,418)(182,417)(183,419)(184,420)
(185,413)(186,414)(187,416)(188,415)(189,410)(190,409)(191,411)(192,412)
(193,405)(194,406)(195,408)(196,407)(197,402)(198,401)(199,403)(200,404)
(201,397)(202,398)(203,400)(204,399)(205,394)(206,393)(207,395)(208,396)
(209,389)(210,390)(211,392)(212,391)(213,386)(214,385)(215,387)(216,388);;
s1 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 31)(  6, 32)(  7, 29)(  8, 30)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 39)( 22, 40)( 23, 37)( 24, 38)
( 49, 65)( 50, 66)( 51, 68)( 52, 67)( 53, 71)( 54, 72)( 55, 69)( 56, 70)
( 59, 60)( 61, 63)( 62, 64)( 73,169)( 74,170)( 75,172)( 76,171)( 77,175)
( 78,176)( 79,173)( 80,174)( 81,185)( 82,186)( 83,188)( 84,187)( 85,191)
( 86,192)( 87,189)( 88,190)( 89,177)( 90,178)( 91,180)( 92,179)( 93,183)
( 94,184)( 95,181)( 96,182)( 97,145)( 98,146)( 99,148)(100,147)(101,151)
(102,152)(103,149)(104,150)(105,161)(106,162)(107,164)(108,163)(109,167)
(110,168)(111,165)(112,166)(113,153)(114,154)(115,156)(116,155)(117,159)
(118,160)(119,157)(120,158)(121,209)(122,210)(123,212)(124,211)(125,215)
(126,216)(127,213)(128,214)(129,201)(130,202)(131,204)(132,203)(133,207)
(134,208)(135,205)(136,206)(137,193)(138,194)(139,196)(140,195)(141,199)
(142,200)(143,197)(144,198)(217,242)(218,241)(219,243)(220,244)(221,248)
(222,247)(223,246)(224,245)(225,258)(226,257)(227,259)(228,260)(229,264)
(230,263)(231,262)(232,261)(233,250)(234,249)(235,251)(236,252)(237,256)
(238,255)(239,254)(240,253)(265,282)(266,281)(267,283)(268,284)(269,288)
(270,287)(271,286)(272,285)(273,274)(277,280)(278,279)(289,386)(290,385)
(291,387)(292,388)(293,392)(294,391)(295,390)(296,389)(297,402)(298,401)
(299,403)(300,404)(301,408)(302,407)(303,406)(304,405)(305,394)(306,393)
(307,395)(308,396)(309,400)(310,399)(311,398)(312,397)(313,362)(314,361)
(315,363)(316,364)(317,368)(318,367)(319,366)(320,365)(321,378)(322,377)
(323,379)(324,380)(325,384)(326,383)(327,382)(328,381)(329,370)(330,369)
(331,371)(332,372)(333,376)(334,375)(335,374)(336,373)(337,426)(338,425)
(339,427)(340,428)(341,432)(342,431)(343,430)(344,429)(345,418)(346,417)
(347,419)(348,420)(349,424)(350,423)(351,422)(352,421)(353,410)(354,409)
(355,411)(356,412)(357,416)(358,415)(359,414)(360,413);;
s2 := (  1, 73)(  2, 74)(  3, 79)(  4, 80)(  5, 78)(  6, 77)(  7, 75)(  8, 76)
(  9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)( 16, 84)
( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)( 24, 92)
( 25, 97)( 26, 98)( 27,103)( 28,104)( 29,102)( 30,101)( 31, 99)( 32,100)
( 33,105)( 34,106)( 35,111)( 36,112)( 37,110)( 38,109)( 39,107)( 40,108)
( 41,113)( 42,114)( 43,119)( 44,120)( 45,118)( 46,117)( 47,115)( 48,116)
( 49,121)( 50,122)( 51,127)( 52,128)( 53,126)( 54,125)( 55,123)( 56,124)
( 57,129)( 58,130)( 59,135)( 60,136)( 61,134)( 62,133)( 63,131)( 64,132)
( 65,137)( 66,138)( 67,143)( 68,144)( 69,142)( 70,141)( 71,139)( 72,140)
(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)(164,168)
(165,166)(171,175)(172,176)(173,174)(179,183)(180,184)(181,182)(187,191)
(188,192)(189,190)(195,199)(196,200)(197,198)(203,207)(204,208)(205,206)
(211,215)(212,216)(213,214)(217,290)(218,289)(219,296)(220,295)(221,293)
(222,294)(223,292)(224,291)(225,298)(226,297)(227,304)(228,303)(229,301)
(230,302)(231,300)(232,299)(233,306)(234,305)(235,312)(236,311)(237,309)
(238,310)(239,308)(240,307)(241,314)(242,313)(243,320)(244,319)(245,317)
(246,318)(247,316)(248,315)(249,322)(250,321)(251,328)(252,327)(253,325)
(254,326)(255,324)(256,323)(257,330)(258,329)(259,336)(260,335)(261,333)
(262,334)(263,332)(264,331)(265,338)(266,337)(267,344)(268,343)(269,341)
(270,342)(271,340)(272,339)(273,346)(274,345)(275,352)(276,351)(277,349)
(278,350)(279,348)(280,347)(281,354)(282,353)(283,360)(284,359)(285,357)
(286,358)(287,356)(288,355)(361,362)(363,368)(364,367)(369,370)(371,376)
(372,375)(377,378)(379,384)(380,383)(385,386)(387,392)(388,391)(393,394)
(395,400)(396,399)(401,402)(403,408)(404,407)(409,410)(411,416)(412,415)
(417,418)(419,424)(420,423)(425,426)(427,432)(428,431);;
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*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope