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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}*1296e
if this polytope has a name.
Group : SmallGroup(1296,891)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 54, 324, 36
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,6}*648a
3-fold quotients : {18,12}*432b, {6,12}*432c
4-fold quotients : {9,6}*324a
6-fold quotients : {18,6}*216b, {6,6}*216c
9-fold quotients : {18,4}*144a, {6,12}*144c
12-fold quotients : {9,6}*108, {3,6}*108
18-fold quotients : {18,2}*72, {6,6}*72c
27-fold quotients : {6,4}*48a
36-fold quotients : {9,2}*36, {3,6}*36
54-fold quotients : {6,2}*24
81-fold quotients : {2,4}*16
108-fold quotients : {3,2}*12
162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 28, 61)( 29, 62)( 30, 63)( 31, 58)
( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 79)( 38, 80)( 39, 81)
( 40, 76)( 41, 77)( 42, 78)( 43, 73)( 44, 74)( 45, 75)( 46, 70)( 47, 71)
( 48, 72)( 49, 67)( 50, 68)( 51, 69)( 52, 64)( 53, 65)( 54, 66)( 85, 88)
( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)
( 97,103)( 98,104)( 99,105)(109,142)(110,143)(111,144)(112,139)(113,140)
(114,141)(115,136)(116,137)(117,138)(118,160)(119,161)(120,162)(121,157)
(122,158)(123,159)(124,154)(125,155)(126,156)(127,151)(128,152)(129,153)
(130,148)(131,149)(132,150)(133,145)(134,146)(135,147)(166,169)(167,170)
(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)
(179,185)(180,186)(190,223)(191,224)(192,225)(193,220)(194,221)(195,222)
(196,217)(197,218)(198,219)(199,241)(200,242)(201,243)(202,238)(203,239)
(204,240)(205,235)(206,236)(207,237)(208,232)(209,233)(210,234)(211,229)
(212,230)(213,231)(214,226)(215,227)(216,228)(247,250)(248,251)(249,252)
(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)(260,266)
(261,267)(271,304)(272,305)(273,306)(274,301)(275,302)(276,303)(277,298)
(278,299)(279,300)(280,322)(281,323)(282,324)(283,319)(284,320)(285,321)
(286,316)(287,317)(288,318)(289,313)(290,314)(291,315)(292,310)(293,311)
(294,312)(295,307)(296,308)(297,309);;
s1 := (  1,227)(  2,228)(  3,226)(  4,233)(  5,234)(  6,232)(  7,230)(  8,231)
(  9,229)( 10,219)( 11,217)( 12,218)( 13,225)( 14,223)( 15,224)( 16,222)
( 17,220)( 18,221)( 19,235)( 20,236)( 21,237)( 22,241)( 23,242)( 24,243)
( 25,238)( 26,239)( 27,240)( 28,200)( 29,201)( 30,199)( 31,206)( 32,207)
( 33,205)( 34,203)( 35,204)( 36,202)( 37,192)( 38,190)( 39,191)( 40,198)
( 41,196)( 42,197)( 43,195)( 44,193)( 45,194)( 46,208)( 47,209)( 48,210)
( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,173)( 56,174)
( 57,172)( 58,179)( 59,180)( 60,178)( 61,176)( 62,177)( 63,175)( 64,165)
( 65,163)( 66,164)( 67,171)( 68,169)( 69,170)( 70,168)( 71,166)( 72,167)
( 73,181)( 74,182)( 75,183)( 76,187)( 77,188)( 78,189)( 79,184)( 80,185)
( 81,186)( 82,308)( 83,309)( 84,307)( 85,314)( 86,315)( 87,313)( 88,311)
( 89,312)( 90,310)( 91,300)( 92,298)( 93,299)( 94,306)( 95,304)( 96,305)
( 97,303)( 98,301)( 99,302)(100,316)(101,317)(102,318)(103,322)(104,323)
(105,324)(106,319)(107,320)(108,321)(109,281)(110,282)(111,280)(112,287)
(113,288)(114,286)(115,284)(116,285)(117,283)(118,273)(119,271)(120,272)
(121,279)(122,277)(123,278)(124,276)(125,274)(126,275)(127,289)(128,290)
(129,291)(130,295)(131,296)(132,297)(133,292)(134,293)(135,294)(136,254)
(137,255)(138,253)(139,260)(140,261)(141,259)(142,257)(143,258)(144,256)
(145,246)(146,244)(147,245)(148,252)(149,250)(150,251)(151,249)(152,247)
(153,248)(154,262)(155,263)(156,264)(157,268)(158,269)(159,270)(160,265)
(161,266)(162,267);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)(170,252)
(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)(178,268)
(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)(186,257)
(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)(194,276)
(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)(202,292)
(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)(210,281)
(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)(218,300)
(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)(226,316)
(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)(234,323)
(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)(242,315)
(243,314);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(324)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 28, 61)( 29, 62)( 30, 63)
( 31, 58)( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 79)( 38, 80)
( 39, 81)( 40, 76)( 41, 77)( 42, 78)( 43, 73)( 44, 74)( 45, 75)( 46, 70)
( 47, 71)( 48, 72)( 49, 67)( 50, 68)( 51, 69)( 52, 64)( 53, 65)( 54, 66)
( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)
( 96,108)( 97,103)( 98,104)( 99,105)(109,142)(110,143)(111,144)(112,139)
(113,140)(114,141)(115,136)(116,137)(117,138)(118,160)(119,161)(120,162)
(121,157)(122,158)(123,159)(124,154)(125,155)(126,156)(127,151)(128,152)
(129,153)(130,148)(131,149)(132,150)(133,145)(134,146)(135,147)(166,169)
(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)
(178,184)(179,185)(180,186)(190,223)(191,224)(192,225)(193,220)(194,221)
(195,222)(196,217)(197,218)(198,219)(199,241)(200,242)(201,243)(202,238)
(203,239)(204,240)(205,235)(206,236)(207,237)(208,232)(209,233)(210,234)
(211,229)(212,230)(213,231)(214,226)(215,227)(216,228)(247,250)(248,251)
(249,252)(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)
(260,266)(261,267)(271,304)(272,305)(273,306)(274,301)(275,302)(276,303)
(277,298)(278,299)(279,300)(280,322)(281,323)(282,324)(283,319)(284,320)
(285,321)(286,316)(287,317)(288,318)(289,313)(290,314)(291,315)(292,310)
(293,311)(294,312)(295,307)(296,308)(297,309);
s1 := Sym(324)!(  1,227)(  2,228)(  3,226)(  4,233)(  5,234)(  6,232)(  7,230)
(  8,231)(  9,229)( 10,219)( 11,217)( 12,218)( 13,225)( 14,223)( 15,224)
( 16,222)( 17,220)( 18,221)( 19,235)( 20,236)( 21,237)( 22,241)( 23,242)
( 24,243)( 25,238)( 26,239)( 27,240)( 28,200)( 29,201)( 30,199)( 31,206)
( 32,207)( 33,205)( 34,203)( 35,204)( 36,202)( 37,192)( 38,190)( 39,191)
( 40,198)( 41,196)( 42,197)( 43,195)( 44,193)( 45,194)( 46,208)( 47,209)
( 48,210)( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,173)
( 56,174)( 57,172)( 58,179)( 59,180)( 60,178)( 61,176)( 62,177)( 63,175)
( 64,165)( 65,163)( 66,164)( 67,171)( 68,169)( 69,170)( 70,168)( 71,166)
( 72,167)( 73,181)( 74,182)( 75,183)( 76,187)( 77,188)( 78,189)( 79,184)
( 80,185)( 81,186)( 82,308)( 83,309)( 84,307)( 85,314)( 86,315)( 87,313)
( 88,311)( 89,312)( 90,310)( 91,300)( 92,298)( 93,299)( 94,306)( 95,304)
( 96,305)( 97,303)( 98,301)( 99,302)(100,316)(101,317)(102,318)(103,322)
(104,323)(105,324)(106,319)(107,320)(108,321)(109,281)(110,282)(111,280)
(112,287)(113,288)(114,286)(115,284)(116,285)(117,283)(118,273)(119,271)
(120,272)(121,279)(122,277)(123,278)(124,276)(125,274)(126,275)(127,289)
(128,290)(129,291)(130,295)(131,296)(132,297)(133,292)(134,293)(135,294)
(136,254)(137,255)(138,253)(139,260)(140,261)(141,259)(142,257)(143,258)
(144,256)(145,246)(146,244)(147,245)(148,252)(149,250)(150,251)(151,249)
(152,247)(153,248)(154,262)(155,263)(156,264)(157,268)(158,269)(159,270)
(160,265)(161,266)(162,267);
s2 := Sym(324)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(163,244)(164,246)(165,245)(166,247)(167,249)(168,248)(169,250)
(170,252)(171,251)(172,262)(173,264)(174,263)(175,265)(176,267)(177,266)
(178,268)(179,270)(180,269)(181,253)(182,255)(183,254)(184,256)(185,258)
(186,257)(187,259)(188,261)(189,260)(190,271)(191,273)(192,272)(193,274)
(194,276)(195,275)(196,277)(197,279)(198,278)(199,289)(200,291)(201,290)
(202,292)(203,294)(204,293)(205,295)(206,297)(207,296)(208,280)(209,282)
(210,281)(211,283)(212,285)(213,284)(214,286)(215,288)(216,287)(217,298)
(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)(225,305)
(226,316)(227,318)(228,317)(229,319)(230,321)(231,320)(232,322)(233,324)
(234,323)(235,307)(236,309)(237,308)(238,310)(239,312)(240,311)(241,313)
(242,315)(243,314);
poly := sub<Sym(324)|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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*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 >;

```
References : None.
to this polytope