Polytope of Type {12,28}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope