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# Polytope of Type {18,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30229)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 72, 432, 48
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
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 : {18,12}*864a
3-fold quotients : {18,4}*576b, {6,12}*576b
4-fold quotients : {18,12}*432a, {18,12}*432c
6-fold quotients : {18,4}*288, {6,12}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {6,4}*192b
12-fold quotients : {18,4}*144a, {9,4}*144, {18,4}*144b, {18,4}*144c, {6,12}*144a, {6,12}*144d
18-fold quotients : {6,4}*96
24-fold quotients : {9,4}*72, {18,2}*72, {6,6}*72a
36-fold quotients : {2,12}*48, {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {9,2}*36
72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*16
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 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 37, 81)
( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 73)
( 46, 74)( 47, 76)( 48, 75)( 49, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)
( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,105)
( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)( 69, 97)
( 70, 98)( 71,100)( 72, 99)(111,112)(113,117)(114,118)(115,120)(116,119)
(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)(138,142)
(139,144)(140,143)(145,189)(146,190)(147,192)(148,191)(149,185)(150,186)
(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)(157,201)(158,202)
(159,204)(160,203)(161,197)(162,198)(163,200)(164,199)(165,193)(166,194)
(167,196)(168,195)(169,213)(170,214)(171,216)(172,215)(173,209)(174,210)
(175,212)(176,211)(177,205)(178,206)(179,208)(180,207)(219,220)(221,225)
(222,226)(223,228)(224,227)(231,232)(233,237)(234,238)(235,240)(236,239)
(243,244)(245,249)(246,250)(247,252)(248,251)(253,297)(254,298)(255,300)
(256,299)(257,293)(258,294)(259,296)(260,295)(261,289)(262,290)(263,292)
(264,291)(265,309)(266,310)(267,312)(268,311)(269,305)(270,306)(271,308)
(272,307)(273,301)(274,302)(275,304)(276,303)(277,321)(278,322)(279,324)
(280,323)(281,317)(282,318)(283,320)(284,319)(285,313)(286,314)(287,316)
(288,315)(327,328)(329,333)(330,334)(331,336)(332,335)(339,340)(341,345)
(342,346)(343,348)(344,347)(351,352)(353,357)(354,358)(355,360)(356,359)
(361,405)(362,406)(363,408)(364,407)(365,401)(366,402)(367,404)(368,403)
(369,397)(370,398)(371,400)(372,399)(373,417)(374,418)(375,420)(376,419)
(377,413)(378,414)(379,416)(380,415)(381,409)(382,410)(383,412)(384,411)
(385,429)(386,430)(387,432)(388,431)(389,425)(390,426)(391,428)(392,427)
(393,421)(394,422)(395,424)(396,423);;
s1 := (  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 45)(  6, 48)(  7, 47)(  8, 46)
(  9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)( 32, 58)
( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 73, 81)( 74, 84)( 75, 83)( 76, 82)
( 78, 80)( 85,105)( 86,108)( 87,107)( 88,106)( 89,101)( 90,104)( 91,103)
( 92,102)( 93, 97)( 94,100)( 95, 99)( 96, 98)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)(127,179)
(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)(135,159)
(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)(143,163)
(144,162)(181,189)(182,192)(183,191)(184,190)(186,188)(193,213)(194,216)
(195,215)(196,214)(197,209)(198,212)(199,211)(200,210)(201,205)(202,208)
(203,207)(204,206)(217,361)(218,364)(219,363)(220,362)(221,369)(222,372)
(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,385)(230,388)
(231,387)(232,386)(233,393)(234,396)(235,395)(236,394)(237,389)(238,392)
(239,391)(240,390)(241,373)(242,376)(243,375)(244,374)(245,381)(246,384)
(247,383)(248,382)(249,377)(250,380)(251,379)(252,378)(253,325)(254,328)
(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)(262,332)
(263,331)(264,330)(265,349)(266,352)(267,351)(268,350)(269,357)(270,360)
(271,359)(272,358)(273,353)(274,356)(275,355)(276,354)(277,337)(278,340)
(279,339)(280,338)(281,345)(282,348)(283,347)(284,346)(285,341)(286,344)
(287,343)(288,342)(289,405)(290,408)(291,407)(292,406)(293,401)(294,404)
(295,403)(296,402)(297,397)(298,400)(299,399)(300,398)(301,429)(302,432)
(303,431)(304,430)(305,425)(306,428)(307,427)(308,426)(309,421)(310,424)
(311,423)(312,422)(313,417)(314,420)(315,419)(316,418)(317,413)(318,416)
(319,415)(320,414)(321,409)(322,412)(323,411)(324,410);;
s2 := (  1,230)(  2,229)(  3,232)(  4,231)(  5,234)(  6,233)(  7,236)(  8,235)
(  9,238)( 10,237)( 11,240)( 12,239)( 13,218)( 14,217)( 15,220)( 16,219)
( 17,222)( 18,221)( 19,224)( 20,223)( 21,226)( 22,225)( 23,228)( 24,227)
( 25,242)( 26,241)( 27,244)( 28,243)( 29,246)( 30,245)( 31,248)( 32,247)
( 33,250)( 34,249)( 35,252)( 36,251)( 37,266)( 38,265)( 39,268)( 40,267)
( 41,270)( 42,269)( 43,272)( 44,271)( 45,274)( 46,273)( 47,276)( 48,275)
( 49,254)( 50,253)( 51,256)( 52,255)( 53,258)( 54,257)( 55,260)( 56,259)
( 57,262)( 58,261)( 59,264)( 60,263)( 61,278)( 62,277)( 63,280)( 64,279)
( 65,282)( 66,281)( 67,284)( 68,283)( 69,286)( 70,285)( 71,288)( 72,287)
( 73,302)( 74,301)( 75,304)( 76,303)( 77,306)( 78,305)( 79,308)( 80,307)
( 81,310)( 82,309)( 83,312)( 84,311)( 85,290)( 86,289)( 87,292)( 88,291)
( 89,294)( 90,293)( 91,296)( 92,295)( 93,298)( 94,297)( 95,300)( 96,299)
( 97,314)( 98,313)( 99,316)(100,315)(101,318)(102,317)(103,320)(104,319)
(105,322)(106,321)(107,324)(108,323)(109,338)(110,337)(111,340)(112,339)
(113,342)(114,341)(115,344)(116,343)(117,346)(118,345)(119,348)(120,347)
(121,326)(122,325)(123,328)(124,327)(125,330)(126,329)(127,332)(128,331)
(129,334)(130,333)(131,336)(132,335)(133,350)(134,349)(135,352)(136,351)
(137,354)(138,353)(139,356)(140,355)(141,358)(142,357)(143,360)(144,359)
(145,374)(146,373)(147,376)(148,375)(149,378)(150,377)(151,380)(152,379)
(153,382)(154,381)(155,384)(156,383)(157,362)(158,361)(159,364)(160,363)
(161,366)(162,365)(163,368)(164,367)(165,370)(166,369)(167,372)(168,371)
(169,386)(170,385)(171,388)(172,387)(173,390)(174,389)(175,392)(176,391)
(177,394)(178,393)(179,396)(180,395)(181,410)(182,409)(183,412)(184,411)
(185,414)(186,413)(187,416)(188,415)(189,418)(190,417)(191,420)(192,419)
(193,398)(194,397)(195,400)(196,399)(197,402)(198,401)(199,404)(200,403)
(201,406)(202,405)(203,408)(204,407)(205,422)(206,421)(207,424)(208,423)
(209,426)(210,425)(211,428)(212,427)(213,430)(214,429)(215,432)(216,431);;
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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 37, 81)( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)
( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49, 93)( 50, 94)( 51, 96)( 52, 95)
( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)
( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)
( 69, 97)( 70, 98)( 71,100)( 72, 99)(111,112)(113,117)(114,118)(115,120)
(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)
(138,142)(139,144)(140,143)(145,189)(146,190)(147,192)(148,191)(149,185)
(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)(157,201)
(158,202)(159,204)(160,203)(161,197)(162,198)(163,200)(164,199)(165,193)
(166,194)(167,196)(168,195)(169,213)(170,214)(171,216)(172,215)(173,209)
(174,210)(175,212)(176,211)(177,205)(178,206)(179,208)(180,207)(219,220)
(221,225)(222,226)(223,228)(224,227)(231,232)(233,237)(234,238)(235,240)
(236,239)(243,244)(245,249)(246,250)(247,252)(248,251)(253,297)(254,298)
(255,300)(256,299)(257,293)(258,294)(259,296)(260,295)(261,289)(262,290)
(263,292)(264,291)(265,309)(266,310)(267,312)(268,311)(269,305)(270,306)
(271,308)(272,307)(273,301)(274,302)(275,304)(276,303)(277,321)(278,322)
(279,324)(280,323)(281,317)(282,318)(283,320)(284,319)(285,313)(286,314)
(287,316)(288,315)(327,328)(329,333)(330,334)(331,336)(332,335)(339,340)
(341,345)(342,346)(343,348)(344,347)(351,352)(353,357)(354,358)(355,360)
(356,359)(361,405)(362,406)(363,408)(364,407)(365,401)(366,402)(367,404)
(368,403)(369,397)(370,398)(371,400)(372,399)(373,417)(374,418)(375,420)
(376,419)(377,413)(378,414)(379,416)(380,415)(381,409)(382,410)(383,412)
(384,411)(385,429)(386,430)(387,432)(388,431)(389,425)(390,426)(391,428)
(392,427)(393,421)(394,422)(395,424)(396,423);
s1 := Sym(432)!(  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 45)(  6, 48)(  7, 47)
(  8, 46)(  9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)
( 24, 66)( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)
( 32, 58)( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 73, 81)( 74, 84)( 75, 83)
( 76, 82)( 78, 80)( 85,105)( 86,108)( 87,107)( 88,106)( 89,101)( 90,104)
( 91,103)( 92,102)( 93, 97)( 94,100)( 95, 99)( 96, 98)(109,145)(110,148)
(111,147)(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)
(119,151)(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)
(127,179)(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)
(135,159)(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)
(143,163)(144,162)(181,189)(182,192)(183,191)(184,190)(186,188)(193,213)
(194,216)(195,215)(196,214)(197,209)(198,212)(199,211)(200,210)(201,205)
(202,208)(203,207)(204,206)(217,361)(218,364)(219,363)(220,362)(221,369)
(222,372)(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,385)
(230,388)(231,387)(232,386)(233,393)(234,396)(235,395)(236,394)(237,389)
(238,392)(239,391)(240,390)(241,373)(242,376)(243,375)(244,374)(245,381)
(246,384)(247,383)(248,382)(249,377)(250,380)(251,379)(252,378)(253,325)
(254,328)(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)
(262,332)(263,331)(264,330)(265,349)(266,352)(267,351)(268,350)(269,357)
(270,360)(271,359)(272,358)(273,353)(274,356)(275,355)(276,354)(277,337)
(278,340)(279,339)(280,338)(281,345)(282,348)(283,347)(284,346)(285,341)
(286,344)(287,343)(288,342)(289,405)(290,408)(291,407)(292,406)(293,401)
(294,404)(295,403)(296,402)(297,397)(298,400)(299,399)(300,398)(301,429)
(302,432)(303,431)(304,430)(305,425)(306,428)(307,427)(308,426)(309,421)
(310,424)(311,423)(312,422)(313,417)(314,420)(315,419)(316,418)(317,413)
(318,416)(319,415)(320,414)(321,409)(322,412)(323,411)(324,410);
s2 := Sym(432)!(  1,230)(  2,229)(  3,232)(  4,231)(  5,234)(  6,233)(  7,236)
(  8,235)(  9,238)( 10,237)( 11,240)( 12,239)( 13,218)( 14,217)( 15,220)
( 16,219)( 17,222)( 18,221)( 19,224)( 20,223)( 21,226)( 22,225)( 23,228)
( 24,227)( 25,242)( 26,241)( 27,244)( 28,243)( 29,246)( 30,245)( 31,248)
( 32,247)( 33,250)( 34,249)( 35,252)( 36,251)( 37,266)( 38,265)( 39,268)
( 40,267)( 41,270)( 42,269)( 43,272)( 44,271)( 45,274)( 46,273)( 47,276)
( 48,275)( 49,254)( 50,253)( 51,256)( 52,255)( 53,258)( 54,257)( 55,260)
( 56,259)( 57,262)( 58,261)( 59,264)( 60,263)( 61,278)( 62,277)( 63,280)
( 64,279)( 65,282)( 66,281)( 67,284)( 68,283)( 69,286)( 70,285)( 71,288)
( 72,287)( 73,302)( 74,301)( 75,304)( 76,303)( 77,306)( 78,305)( 79,308)
( 80,307)( 81,310)( 82,309)( 83,312)( 84,311)( 85,290)( 86,289)( 87,292)
( 88,291)( 89,294)( 90,293)( 91,296)( 92,295)( 93,298)( 94,297)( 95,300)
( 96,299)( 97,314)( 98,313)( 99,316)(100,315)(101,318)(102,317)(103,320)
(104,319)(105,322)(106,321)(107,324)(108,323)(109,338)(110,337)(111,340)
(112,339)(113,342)(114,341)(115,344)(116,343)(117,346)(118,345)(119,348)
(120,347)(121,326)(122,325)(123,328)(124,327)(125,330)(126,329)(127,332)
(128,331)(129,334)(130,333)(131,336)(132,335)(133,350)(134,349)(135,352)
(136,351)(137,354)(138,353)(139,356)(140,355)(141,358)(142,357)(143,360)
(144,359)(145,374)(146,373)(147,376)(148,375)(149,378)(150,377)(151,380)
(152,379)(153,382)(154,381)(155,384)(156,383)(157,362)(158,361)(159,364)
(160,363)(161,366)(162,365)(163,368)(164,367)(165,370)(166,369)(167,372)
(168,371)(169,386)(170,385)(171,388)(172,387)(173,390)(174,389)(175,392)
(176,391)(177,394)(178,393)(179,396)(180,395)(181,410)(182,409)(183,412)
(184,411)(185,414)(186,413)(187,416)(188,415)(189,418)(190,417)(191,420)
(192,419)(193,398)(194,397)(195,400)(196,399)(197,402)(198,401)(199,404)
(200,403)(201,406)(202,405)(203,408)(204,407)(205,422)(206,421)(207,424)
(208,423)(209,426)(210,425)(211,428)(212,427)(213,430)(214,429)(215,432)
(216,431);
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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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