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# Polytope of Type {6,12}

Atlas Canonical Name : {6,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,839)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 54, 324, 108
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 : {6,6}*648b
3-fold quotients : {6,12}*432b
4-fold quotients : {6,6}*324a
6-fold quotients : {6,6}*216b
9-fold quotients : {6,12}*144a
12-fold quotients : {6,6}*108
18-fold quotients : {6,6}*72a
27-fold quotients : {2,12}*48, {6,4}*48a
54-fold quotients : {2,6}*24, {6,2}*24
81-fold quotients : {2,4}*16
108-fold quotients : {2,3}*12, {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,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(190,217)(191,219)(192,218)(193,223)(194,225)(195,224)
(196,220)(197,222)(198,221)(199,226)(200,228)(201,227)(202,232)(203,234)
(204,233)(205,229)(206,231)(207,230)(208,235)(209,237)(210,236)(211,241)
(212,243)(213,242)(214,238)(215,240)(216,239)(245,246)(247,250)(248,252)
(249,251)(254,255)(256,259)(257,261)(258,260)(263,264)(265,268)(266,270)
(267,269)(271,298)(272,300)(273,299)(274,304)(275,306)(276,305)(277,301)
(278,303)(279,302)(280,307)(281,309)(282,308)(283,313)(284,315)(285,314)
(286,310)(287,312)(288,311)(289,316)(290,318)(291,317)(292,322)(293,324)
(294,323)(295,319)(296,321)(297,320);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)
(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)
( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)
( 25, 38)( 26, 37)( 27, 39)( 56, 57)( 58, 60)( 61, 62)( 64, 80)( 65, 79)
( 66, 81)( 67, 73)( 68, 75)( 69, 74)( 70, 78)( 71, 77)( 72, 76)( 82,109)
( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)
( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)
( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)
(107,118)(108,120)(137,138)(139,141)(142,143)(145,161)(146,160)(147,162)
(148,154)(149,156)(150,155)(151,159)(152,158)(153,157)(163,271)(164,273)
(165,272)(166,276)(167,275)(168,274)(169,278)(170,277)(171,279)(172,296)
(173,295)(174,297)(175,289)(176,291)(177,290)(178,294)(179,293)(180,292)
(181,283)(182,285)(183,284)(184,288)(185,287)(186,286)(187,281)(188,280)
(189,282)(190,244)(191,246)(192,245)(193,249)(194,248)(195,247)(196,251)
(197,250)(198,252)(199,269)(200,268)(201,270)(202,262)(203,264)(204,263)
(205,267)(206,266)(207,265)(208,256)(209,258)(210,257)(211,261)(212,260)
(213,259)(214,254)(215,253)(216,255)(217,298)(218,300)(219,299)(220,303)
(221,302)(222,301)(223,305)(224,304)(225,306)(226,323)(227,322)(228,324)
(229,316)(230,318)(231,317)(232,321)(233,320)(234,319)(235,310)(236,312)
(237,311)(238,315)(239,314)(240,313)(241,308)(242,307)(243,309);;
s2 := (  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)( 28,199)( 29,200)( 30,201)( 31,205)( 32,206)
( 33,207)( 34,202)( 35,203)( 36,204)( 37,190)( 38,191)( 39,192)( 40,196)
( 41,197)( 42,198)( 43,193)( 44,194)( 45,195)( 46,208)( 47,209)( 48,210)
( 49,214)( 50,215)( 51,216)( 52,211)( 53,212)( 54,213)( 55,226)( 56,227)
( 57,228)( 58,232)( 59,233)( 60,234)( 61,229)( 62,230)( 63,231)( 64,217)
( 65,218)( 66,219)( 67,223)( 68,224)( 69,225)( 70,220)( 71,221)( 72,222)
( 73,235)( 74,236)( 75,237)( 76,241)( 77,242)( 78,243)( 79,238)( 80,239)
( 81,240)( 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)(109,280)(110,281)(111,282)(112,286)
(113,287)(114,288)(115,283)(116,284)(117,285)(118,271)(119,272)(120,273)
(121,277)(122,278)(123,279)(124,274)(125,275)(126,276)(127,289)(128,290)
(129,291)(130,295)(131,296)(132,297)(133,292)(134,293)(135,294)(136,307)
(137,308)(138,309)(139,313)(140,314)(141,315)(142,310)(143,311)(144,312)
(145,298)(146,299)(147,300)(148,304)(149,305)(150,306)(151,301)(152,302)
(153,303)(154,316)(155,317)(156,318)(157,322)(158,323)(159,324)(160,319)
(161,320)(162,321);;
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*s0*s1*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,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope