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# Polytope of Type {4,24}

Atlas Canonical Name : {4,24}*768h
if this polytope has a name.
Group : SmallGroup(768,1086024)
Rank : 3
Schlafli Type : {4,24}
Number of vertices, edges, etc : 16, 192, 96
Order of s0s1s2 : 24
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-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 : {4,12}*384c
4-fold quotients : {4,6}*192a
8-fold quotients : {4,12}*96c
16-fold quotients : {4,6}*48c
32-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,126)(  2,125)(  3,127)(  4,128)(  5,121)(  6,122)(  7,124)(  8,123)
(  9,118)( 10,117)( 11,119)( 12,120)( 13,113)( 14,114)( 15,116)( 16,115)
( 17,110)( 18,109)( 19,111)( 20,112)( 21,105)( 22,106)( 23,108)( 24,107)
( 25,102)( 26,101)( 27,103)( 28,104)( 29, 97)( 30, 98)( 31,100)( 32, 99)
( 33,158)( 34,157)( 35,159)( 36,160)( 37,153)( 38,154)( 39,156)( 40,155)
( 41,150)( 42,149)( 43,151)( 44,152)( 45,145)( 46,146)( 47,148)( 48,147)
( 49,142)( 50,141)( 51,143)( 52,144)( 53,137)( 54,138)( 55,140)( 56,139)
( 57,134)( 58,133)( 59,135)( 60,136)( 61,129)( 62,130)( 63,132)( 64,131)
( 65,190)( 66,189)( 67,191)( 68,192)( 69,185)( 70,186)( 71,188)( 72,187)
( 73,182)( 74,181)( 75,183)( 76,184)( 77,177)( 78,178)( 79,180)( 80,179)
( 81,174)( 82,173)( 83,175)( 84,176)( 85,169)( 86,170)( 87,172)( 88,171)
( 89,166)( 90,165)( 91,167)( 92,168)( 93,161)( 94,162)( 95,164)( 96,163)
(193,318)(194,317)(195,319)(196,320)(197,313)(198,314)(199,316)(200,315)
(201,310)(202,309)(203,311)(204,312)(205,305)(206,306)(207,308)(208,307)
(209,302)(210,301)(211,303)(212,304)(213,297)(214,298)(215,300)(216,299)
(217,294)(218,293)(219,295)(220,296)(221,289)(222,290)(223,292)(224,291)
(225,350)(226,349)(227,351)(228,352)(229,345)(230,346)(231,348)(232,347)
(233,342)(234,341)(235,343)(236,344)(237,337)(238,338)(239,340)(240,339)
(241,334)(242,333)(243,335)(244,336)(245,329)(246,330)(247,332)(248,331)
(249,326)(250,325)(251,327)(252,328)(253,321)(254,322)(255,324)(256,323)
(257,382)(258,381)(259,383)(260,384)(261,377)(262,378)(263,380)(264,379)
(265,374)(266,373)(267,375)(268,376)(269,369)(270,370)(271,372)(272,371)
(273,366)(274,365)(275,367)(276,368)(277,361)(278,362)(279,364)(280,363)
(281,358)(282,357)(283,359)(284,360)(285,353)(286,354)(287,356)(288,355);;
s1 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)
( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)
( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)
( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)
( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)
( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97, 98)( 99,100)(101,104)(102,103)
(105,106)(107,108)(109,112)(110,111)(113,122)(114,121)(115,124)(116,123)
(117,128)(118,127)(119,126)(120,125)(129,162)(130,161)(131,164)(132,163)
(133,168)(134,167)(135,166)(136,165)(137,170)(138,169)(139,172)(140,171)
(141,176)(142,175)(143,174)(144,173)(145,186)(146,185)(147,188)(148,187)
(149,192)(150,191)(151,190)(152,189)(153,178)(154,177)(155,180)(156,179)
(157,184)(158,183)(159,182)(160,181)(193,289)(194,290)(195,291)(196,292)
(197,295)(198,296)(199,293)(200,294)(201,297)(202,298)(203,299)(204,300)
(205,303)(206,304)(207,301)(208,302)(209,313)(210,314)(211,315)(212,316)
(213,319)(214,320)(215,317)(216,318)(217,305)(218,306)(219,307)(220,308)
(221,311)(222,312)(223,309)(224,310)(225,353)(226,354)(227,355)(228,356)
(229,359)(230,360)(231,357)(232,358)(233,361)(234,362)(235,363)(236,364)
(237,367)(238,368)(239,365)(240,366)(241,377)(242,378)(243,379)(244,380)
(245,383)(246,384)(247,381)(248,382)(249,369)(250,370)(251,371)(252,372)
(253,375)(254,376)(255,373)(256,374)(257,321)(258,322)(259,323)(260,324)
(261,327)(262,328)(263,325)(264,326)(265,329)(266,330)(267,331)(268,332)
(269,335)(270,336)(271,333)(272,334)(273,345)(274,346)(275,347)(276,348)
(277,351)(278,352)(279,349)(280,350)(281,337)(282,338)(283,339)(284,340)
(285,343)(286,344)(287,341)(288,342);;
s2 := (  1,353)(  2,354)(  3,357)(  4,358)(  5,355)(  6,356)(  7,359)(  8,360)
(  9,374)( 10,373)( 11,370)( 12,369)( 13,376)( 14,375)( 15,372)( 16,371)
( 17,364)( 18,363)( 19,368)( 20,367)( 21,362)( 22,361)( 23,366)( 24,365)
( 25,384)( 26,383)( 27,380)( 28,379)( 29,382)( 30,381)( 31,378)( 32,377)
( 33,321)( 34,322)( 35,325)( 36,326)( 37,323)( 38,324)( 39,327)( 40,328)
( 41,342)( 42,341)( 43,338)( 44,337)( 45,344)( 46,343)( 47,340)( 48,339)
( 49,332)( 50,331)( 51,336)( 52,335)( 53,330)( 54,329)( 55,334)( 56,333)
( 57,352)( 58,351)( 59,348)( 60,347)( 61,350)( 62,349)( 63,346)( 64,345)
( 65,289)( 66,290)( 67,293)( 68,294)( 69,291)( 70,292)( 71,295)( 72,296)
( 73,310)( 74,309)( 75,306)( 76,305)( 77,312)( 78,311)( 79,308)( 80,307)
( 81,300)( 82,299)( 83,304)( 84,303)( 85,298)( 86,297)( 87,302)( 88,301)
( 89,320)( 90,319)( 91,316)( 92,315)( 93,318)( 94,317)( 95,314)( 96,313)
( 97,257)( 98,258)( 99,261)(100,262)(101,259)(102,260)(103,263)(104,264)
(105,278)(106,277)(107,274)(108,273)(109,280)(110,279)(111,276)(112,275)
(113,268)(114,267)(115,272)(116,271)(117,266)(118,265)(119,270)(120,269)
(121,288)(122,287)(123,284)(124,283)(125,286)(126,285)(127,282)(128,281)
(129,225)(130,226)(131,229)(132,230)(133,227)(134,228)(135,231)(136,232)
(137,246)(138,245)(139,242)(140,241)(141,248)(142,247)(143,244)(144,243)
(145,236)(146,235)(147,240)(148,239)(149,234)(150,233)(151,238)(152,237)
(153,256)(154,255)(155,252)(156,251)(157,254)(158,253)(159,250)(160,249)
(161,193)(162,194)(163,197)(164,198)(165,195)(166,196)(167,199)(168,200)
(169,214)(170,213)(171,210)(172,209)(173,216)(174,215)(175,212)(176,211)
(177,204)(178,203)(179,208)(180,207)(181,202)(182,201)(183,206)(184,205)
(185,224)(186,223)(187,220)(188,219)(189,222)(190,221)(191,218)(192,217);;
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*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope