Polytope of Type {16,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,3}*768a
if this polytope has a name.
Group : SmallGroup(768,1085768)
Rank : 3
Schlafli Type : {16,3}
Number of vertices, edges, etc : 128, 192, 24
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 16
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {8,3}*384
   4-fold quotients : {8,3}*192
   8-fold quotients : {8,3}*96
   16-fold quotients : {4,3}*48
   32-fold quotients : {4,3}*24
   64-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope