Polytope of Type {28,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,12}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11327)
Rank : 3
Schlafli Type : {28,12}
Number of vertices, edges, etc : 56, 336, 24
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 4
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 : {28,6}*672
   4-fold quotients : {14,12}*336, {28,6}*336b
   7-fold quotients : {4,12}*192b
   8-fold quotients : {14,6}*168
   12-fold quotients : {14,4}*112
   14-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
   24-fold quotients : {14,2}*56
   28-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {7,2}*28
   56-fold quotients : {4,3}*24, {2,6}*24
   84-fold quotients : {2,4}*16
   112-fold quotients : {2,3}*12
   168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope