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# Polytope of Type {12,18}

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

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

```
References : None.
to this polytope