Polytope of Type {12,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,28}*1344a
if this polytope has a name.
Group : SmallGroup(1344,1805)
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 : {12,28}*672
   3-fold quotients : {4,28}*448
   4-fold quotients : {12,14}*336, {6,28}*336a
   6-fold quotients : {4,28}*224
   7-fold quotients : {12,4}*192a
   8-fold quotients : {6,14}*168
   12-fold quotients : {2,28}*112, {4,14}*112
   14-fold quotients : {12,4}*96a
   21-fold quotients : {4,4}*64
   24-fold quotients : {2,14}*56
   28-fold quotients : {12,2}*48, {6,4}*48a
   42-fold quotients : {4,4}*32
   48-fold quotients : {2,7}*28
   56-fold quotients : {6,2}*24
   84-fold quotients : {2,4}*16, {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 := (  8, 15)(  9, 16)( 10, 17)( 11, 18)( 12, 19)( 13, 20)( 14, 21)( 29, 36)
( 30, 37)( 31, 38)( 32, 39)( 33, 40)( 34, 41)( 35, 42)( 43, 64)( 44, 65)
( 45, 66)( 46, 67)( 47, 68)( 48, 69)( 49, 70)( 50, 78)( 51, 79)( 52, 80)
( 53, 81)( 54, 82)( 55, 83)( 56, 84)( 57, 71)( 58, 72)( 59, 73)( 60, 74)
( 61, 75)( 62, 76)( 63, 77)( 92, 99)( 93,100)( 94,101)( 95,102)( 96,103)
( 97,104)( 98,105)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)
(119,126)(127,148)(128,149)(129,150)(130,151)(131,152)(132,153)(133,154)
(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(141,155)
(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(169,253)(170,254)
(171,255)(172,256)(173,257)(174,258)(175,259)(176,267)(177,268)(178,269)
(179,270)(180,271)(181,272)(182,273)(183,260)(184,261)(185,262)(186,263)
(187,264)(188,265)(189,266)(190,274)(191,275)(192,276)(193,277)(194,278)
(195,279)(196,280)(197,288)(198,289)(199,290)(200,291)(201,292)(202,293)
(203,294)(204,281)(205,282)(206,283)(207,284)(208,285)(209,286)(210,287)
(211,316)(212,317)(213,318)(214,319)(215,320)(216,321)(217,322)(218,330)
(219,331)(220,332)(221,333)(222,334)(223,335)(224,336)(225,323)(226,324)
(227,325)(228,326)(229,327)(230,328)(231,329)(232,295)(233,296)(234,297)
(235,298)(236,299)(237,300)(238,301)(239,309)(240,310)(241,311)(242,312)
(243,313)(244,314)(245,315)(246,302)(247,303)(248,304)(249,305)(250,306)
(251,307)(252,308);;
s1 := (  1,176)(  2,182)(  3,181)(  4,180)(  5,179)(  6,178)(  7,177)(  8,169)
(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,183)( 16,189)
( 17,188)( 18,187)( 19,186)( 20,185)( 21,184)( 22,197)( 23,203)( 24,202)
( 25,201)( 26,200)( 27,199)( 28,198)( 29,190)( 30,196)( 31,195)( 32,194)
( 33,193)( 34,192)( 35,191)( 36,204)( 37,210)( 38,209)( 39,208)( 40,207)
( 41,206)( 42,205)( 43,218)( 44,224)( 45,223)( 46,222)( 47,221)( 48,220)
( 49,219)( 50,211)( 51,217)( 52,216)( 53,215)( 54,214)( 55,213)( 56,212)
( 57,225)( 58,231)( 59,230)( 60,229)( 61,228)( 62,227)( 63,226)( 64,239)
( 65,245)( 66,244)( 67,243)( 68,242)( 69,241)( 70,240)( 71,232)( 72,238)
( 73,237)( 74,236)( 75,235)( 76,234)( 77,233)( 78,246)( 79,252)( 80,251)
( 81,250)( 82,249)( 83,248)( 84,247)( 85,260)( 86,266)( 87,265)( 88,264)
( 89,263)( 90,262)( 91,261)( 92,253)( 93,259)( 94,258)( 95,257)( 96,256)
( 97,255)( 98,254)( 99,267)(100,273)(101,272)(102,271)(103,270)(104,269)
(105,268)(106,281)(107,287)(108,286)(109,285)(110,284)(111,283)(112,282)
(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)(120,288)
(121,294)(122,293)(123,292)(124,291)(125,290)(126,289)(127,302)(128,308)
(129,307)(130,306)(131,305)(132,304)(133,303)(134,295)(135,301)(136,300)
(137,299)(138,298)(139,297)(140,296)(141,309)(142,315)(143,314)(144,313)
(145,312)(146,311)(147,310)(148,323)(149,329)(150,328)(151,327)(152,326)
(153,325)(154,324)(155,316)(156,322)(157,321)(158,320)(159,319)(160,318)
(161,317)(162,330)(163,336)(164,335)(165,334)(166,333)(167,332)(168,331);;
s2 := (  1,  2)(  3,  7)(  4,  6)(  8,  9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,107)( 86,106)( 87,112)( 88,111)
( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)( 94,119)( 95,118)( 96,117)
( 97,116)( 98,115)( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)
(105,122)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)(133,150)
(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)(141,163)
(142,162)(143,168)(144,167)(145,166)(146,165)(147,164)(169,212)(170,211)
(171,217)(172,216)(173,215)(174,214)(175,213)(176,219)(177,218)(178,224)
(179,223)(180,222)(181,221)(182,220)(183,226)(184,225)(185,231)(186,230)
(187,229)(188,228)(189,227)(190,233)(191,232)(192,238)(193,237)(194,236)
(195,235)(196,234)(197,240)(198,239)(199,245)(200,244)(201,243)(202,242)
(203,241)(204,247)(205,246)(206,252)(207,251)(208,250)(209,249)(210,248)
(253,317)(254,316)(255,322)(256,321)(257,320)(258,319)(259,318)(260,324)
(261,323)(262,329)(263,328)(264,327)(265,326)(266,325)(267,331)(268,330)
(269,336)(270,335)(271,334)(272,333)(273,332)(274,296)(275,295)(276,301)
(277,300)(278,299)(279,298)(280,297)(281,303)(282,302)(283,308)(284,307)
(285,306)(286,305)(287,304)(288,310)(289,309)(290,315)(291,314)(292,313)
(293,312)(294,311);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope