Polytope of Type {28,12}

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

Permutation Representation (Magma) :

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

References : None.
to this polytope