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# Polytope of Type {66,12}

Atlas Canonical Name : {66,12}*1584c
if this polytope has a name.
Group : SmallGroup(1584,576)
Rank : 3
Schlafli Type : {66,12}
Number of vertices, edges, etc : 66, 396, 12
Order of s0s1s2 : 132
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {66,6}*792c
3-fold quotients : {66,4}*528a
4-fold quotients : {33,6}*396
6-fold quotients : {66,2}*264
9-fold quotients : {22,4}*176
11-fold quotients : {6,12}*144c
12-fold quotients : {33,2}*132
18-fold quotients : {22,2}*88
22-fold quotients : {6,6}*72c
33-fold quotients : {6,4}*48a
36-fold quotients : {11,2}*44
44-fold quotients : {3,6}*36
66-fold quotients : {6,2}*24
99-fold quotients : {2,4}*16
132-fold quotients : {3,2}*12
198-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 12, 23)( 13, 33)( 14, 32)
( 15, 31)( 16, 30)( 17, 29)( 18, 28)( 19, 27)( 20, 26)( 21, 25)( 22, 24)
( 34, 67)( 35, 77)( 36, 76)( 37, 75)( 38, 74)( 39, 73)( 40, 72)( 41, 71)
( 42, 70)( 43, 69)( 44, 68)( 45, 89)( 46, 99)( 47, 98)( 48, 97)( 49, 96)
( 50, 95)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 90)( 56, 78)( 57, 88)
( 58, 87)( 59, 86)( 60, 85)( 61, 84)( 62, 83)( 63, 82)( 64, 81)( 65, 80)
( 66, 79)(101,110)(102,109)(103,108)(104,107)(105,106)(111,122)(112,132)
(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)
(121,123)(133,166)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)
(140,170)(141,169)(142,168)(143,167)(144,188)(145,198)(146,197)(147,196)
(148,195)(149,194)(150,193)(151,192)(152,191)(153,190)(154,189)(155,177)
(156,187)(157,186)(158,185)(159,184)(160,183)(161,182)(162,181)(163,180)
(164,179)(165,178)(200,209)(201,208)(202,207)(203,206)(204,205)(210,221)
(211,231)(212,230)(213,229)(214,228)(215,227)(216,226)(217,225)(218,224)
(219,223)(220,222)(232,265)(233,275)(234,274)(235,273)(236,272)(237,271)
(238,270)(239,269)(240,268)(241,267)(242,266)(243,287)(244,297)(245,296)
(246,295)(247,294)(248,293)(249,292)(250,291)(251,290)(252,289)(253,288)
(254,276)(255,286)(256,285)(257,284)(258,283)(259,282)(260,281)(261,280)
(262,279)(263,278)(264,277)(299,308)(300,307)(301,306)(302,305)(303,304)
(309,320)(310,330)(311,329)(312,328)(313,327)(314,326)(315,325)(316,324)
(317,323)(318,322)(319,321)(331,364)(332,374)(333,373)(334,372)(335,371)
(336,370)(337,369)(338,368)(339,367)(340,366)(341,365)(342,386)(343,396)
(344,395)(345,394)(346,393)(347,392)(348,391)(349,390)(350,389)(351,388)
(352,387)(353,375)(354,385)(355,384)(356,383)(357,382)(358,381)(359,380)
(360,379)(361,378)(362,377)(363,376);;
s1 := (  1,244)(  2,243)(  3,253)(  4,252)(  5,251)(  6,250)(  7,249)(  8,248)
(  9,247)( 10,246)( 11,245)( 12,233)( 13,232)( 14,242)( 15,241)( 16,240)
( 17,239)( 18,238)( 19,237)( 20,236)( 21,235)( 22,234)( 23,255)( 24,254)
( 25,264)( 26,263)( 27,262)( 28,261)( 29,260)( 30,259)( 31,258)( 32,257)
( 33,256)( 34,211)( 35,210)( 36,220)( 37,219)( 38,218)( 39,217)( 40,216)
( 41,215)( 42,214)( 43,213)( 44,212)( 45,200)( 46,199)( 47,209)( 48,208)
( 49,207)( 50,206)( 51,205)( 52,204)( 53,203)( 54,202)( 55,201)( 56,222)
( 57,221)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)( 64,225)
( 65,224)( 66,223)( 67,277)( 68,276)( 69,286)( 70,285)( 71,284)( 72,283)
( 73,282)( 74,281)( 75,280)( 76,279)( 77,278)( 78,266)( 79,265)( 80,275)
( 81,274)( 82,273)( 83,272)( 84,271)( 85,270)( 86,269)( 87,268)( 88,267)
( 89,288)( 90,287)( 91,297)( 92,296)( 93,295)( 94,294)( 95,293)( 96,292)
( 97,291)( 98,290)( 99,289)(100,343)(101,342)(102,352)(103,351)(104,350)
(105,349)(106,348)(107,347)(108,346)(109,345)(110,344)(111,332)(112,331)
(113,341)(114,340)(115,339)(116,338)(117,337)(118,336)(119,335)(120,334)
(121,333)(122,354)(123,353)(124,363)(125,362)(126,361)(127,360)(128,359)
(129,358)(130,357)(131,356)(132,355)(133,310)(134,309)(135,319)(136,318)
(137,317)(138,316)(139,315)(140,314)(141,313)(142,312)(143,311)(144,299)
(145,298)(146,308)(147,307)(148,306)(149,305)(150,304)(151,303)(152,302)
(153,301)(154,300)(155,321)(156,320)(157,330)(158,329)(159,328)(160,327)
(161,326)(162,325)(163,324)(164,323)(165,322)(166,376)(167,375)(168,385)
(169,384)(170,383)(171,382)(172,381)(173,380)(174,379)(175,378)(176,377)
(177,365)(178,364)(179,374)(180,373)(181,372)(182,371)(183,370)(184,369)
(185,368)(186,367)(187,366)(188,387)(189,386)(190,396)(191,395)(192,394)
(193,393)(194,392)(195,391)(196,390)(197,389)(198,388);;
s2 := ( 34, 67)( 35, 68)( 36, 69)( 37, 70)( 38, 71)( 39, 72)( 40, 73)( 41, 74)
( 42, 75)( 43, 76)( 44, 77)( 45, 78)( 46, 79)( 47, 80)( 48, 81)( 49, 82)
( 50, 83)( 51, 84)( 52, 85)( 53, 86)( 54, 87)( 55, 88)( 56, 89)( 57, 90)
( 58, 91)( 59, 92)( 60, 93)( 61, 94)( 62, 95)( 63, 96)( 64, 97)( 65, 98)
( 66, 99)(133,166)(134,167)(135,168)(136,169)(137,170)(138,171)(139,172)
(140,173)(141,174)(142,175)(143,176)(144,177)(145,178)(146,179)(147,180)
(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)(155,188)
(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)(163,196)
(164,197)(165,198)(199,298)(200,299)(201,300)(202,301)(203,302)(204,303)
(205,304)(206,305)(207,306)(208,307)(209,308)(210,309)(211,310)(212,311)
(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)(220,319)
(221,320)(222,321)(223,322)(224,323)(225,324)(226,325)(227,326)(228,327)
(229,328)(230,329)(231,330)(232,364)(233,365)(234,366)(235,367)(236,368)
(237,369)(238,370)(239,371)(240,372)(241,373)(242,374)(243,375)(244,376)
(245,377)(246,378)(247,379)(248,380)(249,381)(250,382)(251,383)(252,384)
(253,385)(254,386)(255,387)(256,388)(257,389)(258,390)(259,391)(260,392)
(261,393)(262,394)(263,395)(264,396)(265,331)(266,332)(267,333)(268,334)
(269,335)(270,336)(271,337)(272,338)(273,339)(274,340)(275,341)(276,342)
(277,343)(278,344)(279,345)(280,346)(281,347)(282,348)(283,349)(284,350)
(285,351)(286,352)(287,353)(288,354)(289,355)(290,356)(291,357)(292,358)
(293,359)(294,360)(295,361)(296,362)(297,363);;
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*s2*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*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*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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope