# Polytope of Type {8,24}

Atlas Canonical Name : {8,24}*768l
if this polytope has a name.
Group : SmallGroup(768,1086641)
Rank : 3
Schlafli Type : {8,24}
Number of vertices, edges, etc : 16, 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}*384f, {4,24}*384d
4-fold quotients : {4,12}*192b, {8,6}*192c
8-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
16-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
32-fold quotients : {4,3}*24, {2,6}*24
48-fold quotients : {2,4}*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 := (  1,199)(  2,200)(  3,197)(  4,198)(  5,196)(  6,195)(  7,194)(  8,193)
(  9,207)( 10,208)( 11,205)( 12,206)( 13,204)( 14,203)( 15,202)( 16,201)
( 17,215)( 18,216)( 19,213)( 20,214)( 21,212)( 22,211)( 23,210)( 24,209)
( 25,223)( 26,224)( 27,221)( 28,222)( 29,220)( 30,219)( 31,218)( 32,217)
( 33,231)( 34,232)( 35,229)( 36,230)( 37,228)( 38,227)( 39,226)( 40,225)
( 41,239)( 42,240)( 43,237)( 44,238)( 45,236)( 46,235)( 47,234)( 48,233)
( 49,247)( 50,248)( 51,245)( 52,246)( 53,244)( 54,243)( 55,242)( 56,241)
( 57,255)( 58,256)( 59,253)( 60,254)( 61,252)( 62,251)( 63,250)( 64,249)
( 65,263)( 66,264)( 67,261)( 68,262)( 69,260)( 70,259)( 71,258)( 72,257)
( 73,271)( 74,272)( 75,269)( 76,270)( 77,268)( 78,267)( 79,266)( 80,265)
( 81,279)( 82,280)( 83,277)( 84,278)( 85,276)( 86,275)( 87,274)( 88,273)
( 89,287)( 90,288)( 91,285)( 92,286)( 93,284)( 94,283)( 95,282)( 96,281)
( 97,320)( 98,319)( 99,318)(100,317)(101,315)(102,316)(103,313)(104,314)
(105,328)(106,327)(107,326)(108,325)(109,323)(110,324)(111,321)(112,322)
(113,336)(114,335)(115,334)(116,333)(117,331)(118,332)(119,329)(120,330)
(121,296)(122,295)(123,294)(124,293)(125,291)(126,292)(127,289)(128,290)
(129,304)(130,303)(131,302)(132,301)(133,299)(134,300)(135,297)(136,298)
(137,312)(138,311)(139,310)(140,309)(141,307)(142,308)(143,305)(144,306)
(145,368)(146,367)(147,366)(148,365)(149,363)(150,364)(151,361)(152,362)
(153,376)(154,375)(155,374)(156,373)(157,371)(158,372)(159,369)(160,370)
(161,384)(162,383)(163,382)(164,381)(165,379)(166,380)(167,377)(168,378)
(169,344)(170,343)(171,342)(172,341)(173,339)(174,340)(175,337)(176,338)
(177,352)(178,351)(179,350)(180,349)(181,347)(182,348)(183,345)(184,346)
(185,360)(186,359)(187,358)(188,357)(189,355)(190,356)(191,353)(192,354);;
s1 := (  1,361)(  2,362)(  3,364)(  4,363)(  5,367)(  6,368)(  7,365)(  8,366)
(  9,377)( 10,378)( 11,380)( 12,379)( 13,383)( 14,384)( 15,381)( 16,382)
( 17,369)( 18,370)( 19,372)( 20,371)( 21,375)( 22,376)( 23,373)( 24,374)
( 25,337)( 26,338)( 27,340)( 28,339)( 29,343)( 30,344)( 31,341)( 32,342)
( 33,353)( 34,354)( 35,356)( 36,355)( 37,359)( 38,360)( 39,357)( 40,358)
( 41,345)( 42,346)( 43,348)( 44,347)( 45,351)( 46,352)( 47,349)( 48,350)
( 49,313)( 50,314)( 51,316)( 52,315)( 53,319)( 54,320)( 55,317)( 56,318)
( 57,329)( 58,330)( 59,332)( 60,331)( 61,335)( 62,336)( 63,333)( 64,334)
( 65,321)( 66,322)( 67,324)( 68,323)( 69,327)( 70,328)( 71,325)( 72,326)
( 73,289)( 74,290)( 75,292)( 76,291)( 77,295)( 78,296)( 79,293)( 80,294)
( 81,305)( 82,306)( 83,308)( 84,307)( 85,311)( 86,312)( 87,309)( 88,310)
( 89,297)( 90,298)( 91,300)( 92,299)( 93,303)( 94,304)( 95,301)( 96,302)
( 97,265)( 98,266)( 99,268)(100,267)(101,271)(102,272)(103,269)(104,270)
(105,281)(106,282)(107,284)(108,283)(109,287)(110,288)(111,285)(112,286)
(113,273)(114,274)(115,276)(116,275)(117,279)(118,280)(119,277)(120,278)
(121,241)(122,242)(123,244)(124,243)(125,247)(126,248)(127,245)(128,246)
(129,257)(130,258)(131,260)(132,259)(133,263)(134,264)(135,261)(136,262)
(137,249)(138,250)(139,252)(140,251)(141,255)(142,256)(143,253)(144,254)
(145,217)(146,218)(147,220)(148,219)(149,223)(150,224)(151,221)(152,222)
(153,233)(154,234)(155,236)(156,235)(157,239)(158,240)(159,237)(160,238)
(161,225)(162,226)(163,228)(164,227)(165,231)(166,232)(167,229)(168,230)
(169,193)(170,194)(171,196)(172,195)(173,199)(174,200)(175,197)(176,198)
(177,209)(178,210)(179,212)(180,211)(181,215)(182,216)(183,213)(184,214)
(185,201)(186,202)(187,204)(188,203)(189,207)(190,208)(191,205)(192,206);;
s2 := (  1, 17)(  2, 18)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 24)(  8, 23)
( 11, 14)( 12, 13)( 15, 16)( 25, 41)( 26, 42)( 27, 46)( 28, 45)( 29, 44)
( 30, 43)( 31, 48)( 32, 47)( 35, 38)( 36, 37)( 39, 40)( 49, 89)( 50, 90)
( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 81)( 58, 82)
( 59, 86)( 60, 85)( 61, 84)( 62, 83)( 63, 88)( 64, 87)( 65, 73)( 66, 74)
( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 80)( 72, 79)( 97,161)( 98,162)
( 99,166)(100,165)(101,164)(102,163)(103,168)(104,167)(105,153)(106,154)
(107,158)(108,157)(109,156)(110,155)(111,160)(112,159)(113,145)(114,146)
(115,150)(116,149)(117,148)(118,147)(119,152)(120,151)(121,185)(122,186)
(123,190)(124,189)(125,188)(126,187)(127,192)(128,191)(129,177)(130,178)
(131,182)(132,181)(133,180)(134,179)(135,184)(136,183)(137,169)(138,170)
(139,174)(140,173)(141,172)(142,171)(143,176)(144,175)(193,210)(194,209)
(195,213)(196,214)(197,211)(198,212)(199,215)(200,216)(201,202)(203,205)
(204,206)(217,234)(218,233)(219,237)(220,238)(221,235)(222,236)(223,239)
(224,240)(225,226)(227,229)(228,230)(241,282)(242,281)(243,285)(244,286)
(245,283)(246,284)(247,287)(248,288)(249,274)(250,273)(251,277)(252,278)
(253,275)(254,276)(255,279)(256,280)(257,266)(258,265)(259,269)(260,270)
(261,267)(262,268)(263,271)(264,272)(289,354)(290,353)(291,357)(292,358)
(293,355)(294,356)(295,359)(296,360)(297,346)(298,345)(299,349)(300,350)
(301,347)(302,348)(303,351)(304,352)(305,338)(306,337)(307,341)(308,342)
(309,339)(310,340)(311,343)(312,344)(313,378)(314,377)(315,381)(316,382)
(317,379)(318,380)(319,383)(320,384)(321,370)(322,369)(323,373)(324,374)
(325,371)(326,372)(327,375)(328,376)(329,362)(330,361)(331,365)(332,366)
(333,363)(334,364)(335,367)(336,368);;
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*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope