Polytope of Type {3,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,16}*768a
if this polytope has a name.
Group : SmallGroup(768,1085768)
Rank : 3
Schlafli Type : {3,16}
Number of vertices, edges, etc : 24, 192, 128
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 16
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 : {3,8}*384
   4-fold quotients : {3,8}*192
   8-fold quotients : {3,8}*96
   16-fold quotients : {3,4}*48
   32-fold quotients : {3,4}*24
   64-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope