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# Polytope of Type {18,12}

Atlas Canonical Name : {18,12}*1296f
if this polytope has a name.
Group : SmallGroup(1296,917)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 54, 324, 36
Order of s0s1s2 : 12
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}*648c
3-fold quotients : {6,12}*432c
4-fold quotients : {9,6}*324c
6-fold quotients : {6,6}*216c
9-fold quotients : {6,12}*144c
12-fold quotients : {3,6}*108
18-fold quotients : {6,6}*72c
27-fold quotients : {6,4}*48a
36-fold quotients : {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 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 73)( 38, 75)( 39, 74)
( 40, 78)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 81)( 46, 64)( 47, 66)
( 48, 65)( 49, 69)( 50, 68)( 51, 67)( 52, 71)( 53, 70)( 54, 72)( 83, 84)
( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)
( 97,107)( 98,106)( 99,108)(109,136)(110,138)(111,137)(112,141)(113,140)
(114,139)(115,143)(116,142)(117,144)(118,154)(119,156)(120,155)(121,159)
(122,158)(123,157)(124,161)(125,160)(126,162)(127,145)(128,147)(129,146)
(130,150)(131,149)(132,148)(133,152)(134,151)(135,153)(164,165)(166,168)
(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)
(179,187)(180,189)(190,217)(191,219)(192,218)(193,222)(194,221)(195,220)
(196,224)(197,223)(198,225)(199,235)(200,237)(201,236)(202,240)(203,239)
(204,238)(205,242)(206,241)(207,243)(208,226)(209,228)(210,227)(211,231)
(212,230)(213,229)(214,233)(215,232)(216,234)(245,246)(247,249)(250,251)
(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)
(261,270)(271,298)(272,300)(273,299)(274,303)(275,302)(276,301)(277,305)
(278,304)(279,306)(280,316)(281,318)(282,317)(283,321)(284,320)(285,319)
(286,323)(287,322)(288,324)(289,307)(290,309)(291,308)(292,312)(293,311)
(294,310)(295,314)(296,313)(297,315);;
s1 := (  1,229)(  2,231)(  3,230)(  4,232)(  5,234)(  6,233)(  7,226)(  8,228)
(  9,227)( 10,223)( 11,225)( 12,224)( 13,217)( 14,219)( 15,218)( 16,220)
( 17,222)( 18,221)( 19,237)( 20,236)( 21,235)( 22,240)( 23,239)( 24,238)
( 25,243)( 26,242)( 27,241)( 28,202)( 29,204)( 30,203)( 31,205)( 32,207)
( 33,206)( 34,199)( 35,201)( 36,200)( 37,196)( 38,198)( 39,197)( 40,190)
( 41,192)( 42,191)( 43,193)( 44,195)( 45,194)( 46,210)( 47,209)( 48,208)
( 49,213)( 50,212)( 51,211)( 52,216)( 53,215)( 54,214)( 55,175)( 56,177)
( 57,176)( 58,178)( 59,180)( 60,179)( 61,172)( 62,174)( 63,173)( 64,169)
( 65,171)( 66,170)( 67,163)( 68,165)( 69,164)( 70,166)( 71,168)( 72,167)
( 73,183)( 74,182)( 75,181)( 76,186)( 77,185)( 78,184)( 79,189)( 80,188)
( 81,187)( 82,310)( 83,312)( 84,311)( 85,313)( 86,315)( 87,314)( 88,307)
( 89,309)( 90,308)( 91,304)( 92,306)( 93,305)( 94,298)( 95,300)( 96,299)
( 97,301)( 98,303)( 99,302)(100,318)(101,317)(102,316)(103,321)(104,320)
(105,319)(106,324)(107,323)(108,322)(109,283)(110,285)(111,284)(112,286)
(113,288)(114,287)(115,280)(116,282)(117,281)(118,277)(119,279)(120,278)
(121,271)(122,273)(123,272)(124,274)(125,276)(126,275)(127,291)(128,290)
(129,289)(130,294)(131,293)(132,292)(133,297)(134,296)(135,295)(136,256)
(137,258)(138,257)(139,259)(140,261)(141,260)(142,253)(143,255)(144,254)
(145,250)(146,252)(147,251)(148,244)(149,246)(150,245)(151,247)(152,249)
(153,248)(154,264)(155,263)(156,262)(157,267)(158,266)(159,265)(160,270)
(161,269)(162,268);;
s2 := (  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)
( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)
(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)
(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)
(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158)
(163,244)(164,245)(165,246)(166,251)(167,252)(168,250)(169,249)(170,247)
(171,248)(172,262)(173,263)(174,264)(175,269)(176,270)(177,268)(178,267)
(179,265)(180,266)(181,253)(182,254)(183,255)(184,260)(185,261)(186,259)
(187,258)(188,256)(189,257)(190,271)(191,272)(192,273)(193,278)(194,279)
(195,277)(196,276)(197,274)(198,275)(199,289)(200,290)(201,291)(202,296)
(203,297)(204,295)(205,294)(206,292)(207,293)(208,280)(209,281)(210,282)
(211,287)(212,288)(213,286)(214,285)(215,283)(216,284)(217,298)(218,299)
(219,300)(220,305)(221,306)(222,304)(223,303)(224,301)(225,302)(226,316)
(227,317)(228,318)(229,323)(230,324)(231,322)(232,321)(233,319)(234,320)
(235,307)(236,308)(237,309)(238,314)(239,315)(240,313)(241,312)(242,310)
(243,311);;
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,
s0*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope