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# Polytope of Type {8,24}

Atlas Canonical Name : {8,24}*768e
if this polytope has a name.
Group : SmallGroup(768,90301)
Rank : 3
Schlafli Type : {8,24}
Number of vertices, edges, etc : 16, 192, 48
Order of s0s1s2 : 12
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 : {4,24}*384b
3-fold quotients : {8,8}*256e
4-fold quotients : {4,12}*192a
6-fold quotients : {4,8}*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 := (  1,193)(  2,194)(  3,195)(  4,196)(  5,197)(  6,198)(  7,202)(  8,203)
(  9,204)( 10,199)( 11,200)( 12,201)( 13,205)( 14,206)( 15,207)( 16,208)
( 17,209)( 18,210)( 19,214)( 20,215)( 21,216)( 22,211)( 23,212)( 24,213)
( 25,238)( 26,239)( 27,240)( 28,235)( 29,236)( 30,237)( 31,229)( 32,230)
( 33,231)( 34,232)( 35,233)( 36,234)( 37,223)( 38,224)( 39,225)( 40,226)
( 41,227)( 42,228)( 43,220)( 44,221)( 45,222)( 46,217)( 47,218)( 48,219)
( 49,244)( 50,245)( 51,246)( 52,241)( 53,242)( 54,243)( 55,247)( 56,248)
( 57,249)( 58,250)( 59,251)( 60,252)( 61,256)( 62,257)( 63,258)( 64,253)
( 65,254)( 66,255)( 67,259)( 68,260)( 69,261)( 70,262)( 71,263)( 72,264)
( 73,283)( 74,284)( 75,285)( 76,286)( 77,287)( 78,288)( 79,280)( 80,281)
( 81,282)( 82,277)( 83,278)( 84,279)( 85,274)( 86,275)( 87,276)( 88,271)
( 89,272)( 90,273)( 91,265)( 92,266)( 93,267)( 94,268)( 95,269)( 96,270)
( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,298)(104,299)
(105,300)(106,295)(107,296)(108,297)(109,301)(110,302)(111,303)(112,304)
(113,305)(114,306)(115,310)(116,311)(117,312)(118,307)(119,308)(120,309)
(121,334)(122,335)(123,336)(124,331)(125,332)(126,333)(127,325)(128,326)
(129,327)(130,328)(131,329)(132,330)(133,319)(134,320)(135,321)(136,322)
(137,323)(138,324)(139,316)(140,317)(141,318)(142,313)(143,314)(144,315)
(145,340)(146,341)(147,342)(148,337)(149,338)(150,339)(151,343)(152,344)
(153,345)(154,346)(155,347)(156,348)(157,352)(158,353)(159,354)(160,349)
(161,350)(162,351)(163,355)(164,356)(165,357)(166,358)(167,359)(168,360)
(169,379)(170,380)(171,381)(172,382)(173,383)(174,384)(175,376)(176,377)
(177,378)(178,373)(179,374)(180,375)(181,370)(182,371)(183,372)(184,367)
(185,368)(186,369)(187,361)(188,362)(189,363)(190,364)(191,365)(192,366);;
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 := (  1,291)(  2,290)(  3,289)(  4,294)(  5,293)(  6,292)(  7,297)(  8,296)
(  9,295)( 10,300)( 11,299)( 12,298)( 13,306)( 14,305)( 15,304)( 16,303)
( 17,302)( 18,301)( 19,312)( 20,311)( 21,310)( 22,309)( 23,308)( 24,307)
( 25,336)( 26,335)( 27,334)( 28,333)( 29,332)( 30,331)( 31,330)( 32,329)
( 33,328)( 34,327)( 35,326)( 36,325)( 37,324)( 38,323)( 39,322)( 40,321)
( 41,320)( 42,319)( 43,318)( 44,317)( 45,316)( 46,315)( 47,314)( 48,313)
( 49,354)( 50,353)( 51,352)( 52,351)( 53,350)( 54,349)( 55,360)( 56,359)
( 57,358)( 58,357)( 59,356)( 60,355)( 61,342)( 62,341)( 63,340)( 64,339)
( 65,338)( 66,337)( 67,348)( 68,347)( 69,346)( 70,345)( 71,344)( 72,343)
( 73,369)( 74,368)( 75,367)( 76,372)( 77,371)( 78,370)( 79,363)( 80,362)
( 81,361)( 82,366)( 83,365)( 84,364)( 85,384)( 86,383)( 87,382)( 88,381)
( 89,380)( 90,379)( 91,378)( 92,377)( 93,376)( 94,375)( 95,374)( 96,373)
( 97,195)( 98,194)( 99,193)(100,198)(101,197)(102,196)(103,201)(104,200)
(105,199)(106,204)(107,203)(108,202)(109,210)(110,209)(111,208)(112,207)
(113,206)(114,205)(115,216)(116,215)(117,214)(118,213)(119,212)(120,211)
(121,240)(122,239)(123,238)(124,237)(125,236)(126,235)(127,234)(128,233)
(129,232)(130,231)(131,230)(132,229)(133,228)(134,227)(135,226)(136,225)
(137,224)(138,223)(139,222)(140,221)(141,220)(142,219)(143,218)(144,217)
(145,258)(146,257)(147,256)(148,255)(149,254)(150,253)(151,264)(152,263)
(153,262)(154,261)(155,260)(156,259)(157,246)(158,245)(159,244)(160,243)
(161,242)(162,241)(163,252)(164,251)(165,250)(166,249)(167,248)(168,247)
(169,273)(170,272)(171,271)(172,276)(173,275)(174,274)(175,267)(176,266)
(177,265)(178,270)(179,269)(180,268)(181,288)(182,287)(183,286)(184,285)
(185,284)(186,283)(187,282)(188,281)(189,280)(190,279)(191,278)(192,277);;
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*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0,
s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s0*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s0*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope