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# Polytope of Type {8,12}

Atlas Canonical Name : {8,12}*768d
if this polytope has a name.
Group : SmallGroup(768,90303)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 32, 192, 48
Order of s0s1s2 : 24
Order of s0s1s2s1 : 8
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 : {8,12}*384b
3-fold quotients : {8,4}*256d
4-fold quotients : {4,12}*192a
6-fold quotients : {8,4}*128b
8-fold quotients : {4,12}*96a
12-fold quotients : {4,4}*64
16-fold quotients : {2,12}*48, {4,6}*48a
24-fold quotients : {4,4}*32
32-fold quotients : {2,6}*24
48-fold quotients : {2,4}*16, {4,2}*16
64-fold quotients : {2,3}*12
96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,289)(  2,290)(  3,291)(  4,292)(  5,293)(  6,294)(  7,295)(  8,296)
(  9,297)( 10,298)( 11,299)( 12,300)( 13,304)( 14,305)( 15,306)( 16,301)
( 17,302)( 18,303)( 19,310)( 20,311)( 21,312)( 22,307)( 23,308)( 24,309)
( 25,334)( 26,335)( 27,336)( 28,331)( 29,332)( 30,333)( 31,328)( 32,329)
( 33,330)( 34,325)( 35,326)( 36,327)( 37,322)( 38,323)( 39,324)( 40,319)
( 41,320)( 42,321)( 43,316)( 44,317)( 45,318)( 46,313)( 47,314)( 48,315)
( 49,352)( 50,353)( 51,354)( 52,349)( 53,350)( 54,351)( 55,358)( 56,359)
( 57,360)( 58,355)( 59,356)( 60,357)( 61,340)( 62,341)( 63,342)( 64,337)
( 65,338)( 66,339)( 67,346)( 68,347)( 69,348)( 70,343)( 71,344)( 72,345)
( 73,367)( 74,368)( 75,369)( 76,370)( 77,371)( 78,372)( 79,361)( 80,362)
( 81,363)( 82,364)( 83,365)( 84,366)( 85,382)( 86,383)( 87,384)( 88,379)
( 89,380)( 90,381)( 91,376)( 92,377)( 93,378)( 94,373)( 95,374)( 96,375)
( 97,193)( 98,194)( 99,195)(100,196)(101,197)(102,198)(103,199)(104,200)
(105,201)(106,202)(107,203)(108,204)(109,208)(110,209)(111,210)(112,205)
(113,206)(114,207)(115,214)(116,215)(117,216)(118,211)(119,212)(120,213)
(121,238)(122,239)(123,240)(124,235)(125,236)(126,237)(127,232)(128,233)
(129,234)(130,229)(131,230)(132,231)(133,226)(134,227)(135,228)(136,223)
(137,224)(138,225)(139,220)(140,221)(141,222)(142,217)(143,218)(144,219)
(145,256)(146,257)(147,258)(148,253)(149,254)(150,255)(151,262)(152,263)
(153,264)(154,259)(155,260)(156,261)(157,244)(158,245)(159,246)(160,241)
(161,242)(162,243)(163,250)(164,251)(165,252)(166,247)(167,248)(168,249)
(169,271)(170,272)(171,273)(172,274)(173,275)(174,276)(175,265)(176,266)
(177,267)(178,268)(179,269)(180,270)(181,286)(182,287)(183,288)(184,283)
(185,284)(186,285)(187,280)(188,281)(189,282)(190,277)(191,278)(192,279);;
s1 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 13, 19)( 14, 21)( 15, 20)( 16, 22)
( 17, 24)( 18, 23)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 37, 43)( 38, 45)
( 39, 44)( 40, 46)( 41, 48)( 42, 47)( 49, 52)( 50, 54)( 51, 53)( 55, 58)
( 56, 60)( 57, 59)( 61, 70)( 62, 72)( 63, 71)( 64, 67)( 65, 69)( 66, 68)
( 73, 76)( 74, 78)( 75, 77)( 79, 82)( 80, 84)( 81, 83)( 85, 94)( 86, 96)
( 87, 95)( 88, 91)( 89, 93)( 90, 92)( 97,121)( 98,123)( 99,122)(100,124)
(101,126)(102,125)(103,127)(104,129)(105,128)(106,130)(107,132)(108,131)
(109,139)(110,141)(111,140)(112,142)(113,144)(114,143)(115,133)(116,135)
(117,134)(118,136)(119,138)(120,137)(145,175)(146,177)(147,176)(148,178)
(149,180)(150,179)(151,169)(152,171)(153,170)(154,172)(155,174)(156,173)
(157,181)(158,183)(159,182)(160,184)(161,186)(162,185)(163,187)(164,189)
(165,188)(166,190)(167,192)(168,191)(193,241)(194,243)(195,242)(196,244)
(197,246)(198,245)(199,247)(200,249)(201,248)(202,250)(203,252)(204,251)
(205,259)(206,261)(207,260)(208,262)(209,264)(210,263)(211,253)(212,255)
(213,254)(214,256)(215,258)(216,257)(217,265)(218,267)(219,266)(220,268)
(221,270)(222,269)(223,271)(224,273)(225,272)(226,274)(227,276)(228,275)
(229,283)(230,285)(231,284)(232,286)(233,288)(234,287)(235,277)(236,279)
(237,278)(238,280)(239,282)(240,281)(289,376)(290,378)(291,377)(292,373)
(293,375)(294,374)(295,382)(296,384)(297,383)(298,379)(299,381)(300,380)
(301,367)(302,369)(303,368)(304,370)(305,372)(306,371)(307,361)(308,363)
(309,362)(310,364)(311,366)(312,365)(313,349)(314,351)(315,350)(316,352)
(317,354)(318,353)(319,355)(320,357)(321,356)(322,358)(323,360)(324,359)
(325,346)(326,348)(327,347)(328,343)(329,345)(330,344)(331,340)(332,342)
(333,341)(334,337)(335,339)(336,338);;
s2 := (  1, 99)(  2, 98)(  3, 97)(  4,102)(  5,101)(  6,100)(  7,108)(  8,107)
(  9,106)( 10,105)( 11,104)( 12,103)( 13,114)( 14,113)( 15,112)( 16,111)
( 17,110)( 18,109)( 19,117)( 20,116)( 21,115)( 22,120)( 23,119)( 24,118)
( 25,123)( 26,122)( 27,121)( 28,126)( 29,125)( 30,124)( 31,132)( 32,131)
( 33,130)( 34,129)( 35,128)( 36,127)( 37,138)( 38,137)( 39,136)( 40,135)
( 41,134)( 42,133)( 43,141)( 44,140)( 45,139)( 46,144)( 47,143)( 48,142)
( 49,159)( 50,158)( 51,157)( 52,162)( 53,161)( 54,160)( 55,168)( 56,167)
( 57,166)( 58,165)( 59,164)( 60,163)( 61,147)( 62,146)( 63,145)( 64,150)
( 65,149)( 66,148)( 67,156)( 68,155)( 69,154)( 70,153)( 71,152)( 72,151)
( 73,186)( 74,185)( 75,184)( 76,183)( 77,182)( 78,181)( 79,189)( 80,188)
( 81,187)( 82,192)( 83,191)( 84,190)( 85,174)( 86,173)( 87,172)( 88,171)
( 89,170)( 90,169)( 91,177)( 92,176)( 93,175)( 94,180)( 95,179)( 96,178)
(193,291)(194,290)(195,289)(196,294)(197,293)(198,292)(199,300)(200,299)
(201,298)(202,297)(203,296)(204,295)(205,306)(206,305)(207,304)(208,303)
(209,302)(210,301)(211,309)(212,308)(213,307)(214,312)(215,311)(216,310)
(217,315)(218,314)(219,313)(220,318)(221,317)(222,316)(223,324)(224,323)
(225,322)(226,321)(227,320)(228,319)(229,330)(230,329)(231,328)(232,327)
(233,326)(234,325)(235,333)(236,332)(237,331)(238,336)(239,335)(240,334)
(241,351)(242,350)(243,349)(244,354)(245,353)(246,352)(247,360)(248,359)
(249,358)(250,357)(251,356)(252,355)(253,339)(254,338)(255,337)(256,342)
(257,341)(258,340)(259,348)(260,347)(261,346)(262,345)(263,344)(264,343)
(265,378)(266,377)(267,376)(268,375)(269,374)(270,373)(271,381)(272,380)
(273,379)(274,384)(275,383)(276,382)(277,366)(278,365)(279,364)(280,363)
(281,362)(282,361)(283,369)(284,368)(285,367)(286,372)(287,371)(288,370);;
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*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope