Polytope of Type {28,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,24}*1344a
Also Known As : {28,24|2}. if this polytope has another name.
Group : SmallGroup(1344,2773)
Rank : 3
Schlafli Type : {28,24}
Number of vertices, edges, etc : 28, 336, 24
Order of s0s1s2 : 168
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {14,24}*672, {28,12}*672
3-fold quotients : {28,8}*448a
4-fold quotients : {14,12}*336, {28,6}*336a
6-fold quotients : {28,4}*224, {14,8}*224
7-fold quotients : {4,24}*192a
8-fold quotients : {14,6}*168
12-fold quotients : {28,2}*112, {14,4}*112
14-fold quotients : {4,12}*96a, {2,24}*96
21-fold quotients : {4,8}*64a
24-fold quotients : {14,2}*56
28-fold quotients : {2,12}*48, {4,6}*48a
42-fold quotients : {4,4}*32, {2,8}*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)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)
(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166)
(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)(253,295)(254,301)(255,300)(256,299)(257,298)(258,297)
(259,296)(260,302)(261,308)(262,307)(263,306)(264,305)(265,304)(266,303)
(267,309)(268,315)(269,314)(270,313)(271,312)(272,311)(273,310)(274,316)
(275,322)(276,321)(277,320)(278,319)(279,318)(280,317)(281,323)(282,329)
(283,328)(284,327)(285,326)(286,325)(287,324)(288,330)(289,336)(290,335)
(291,334)(292,333)(293,332)(294,331);;
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)( 85,275)( 86,274)( 87,280)( 88,279)
( 89,278)( 90,277)( 91,276)( 92,289)( 93,288)( 94,294)( 95,293)( 96,292)
( 97,291)( 98,290)( 99,282)(100,281)(101,287)(102,286)(103,285)(104,284)
(105,283)(106,254)(107,253)(108,259)(109,258)(110,257)(111,256)(112,255)
(113,268)(114,267)(115,273)(116,272)(117,271)(118,270)(119,269)(120,261)
(121,260)(122,266)(123,265)(124,264)(125,263)(126,262)(127,317)(128,316)
(129,322)(130,321)(131,320)(132,319)(133,318)(134,331)(135,330)(136,336)
(137,335)(138,334)(139,333)(140,332)(141,324)(142,323)(143,329)(144,328)
(145,327)(146,326)(147,325)(148,296)(149,295)(150,301)(151,300)(152,299)
(153,298)(154,297)(155,310)(156,309)(157,315)(158,314)(159,313)(160,312)
(161,311)(162,303)(163,302)(164,308)(165,307)(166,306)(167,305)(168,304);;
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, 50)( 44, 51)
( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)( 66, 73)
( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85,113)( 86,114)( 87,115)( 88,116)
( 89,117)( 90,118)( 91,119)( 92,106)( 93,107)( 94,108)( 95,109)( 96,110)
( 97,111)( 98,112)( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)
(105,126)(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)
(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)(218,295)
(219,296)(220,297)(221,298)(222,299)(223,300)(224,301)(225,309)(226,310)
(227,311)(228,312)(229,313)(230,314)(231,315)(232,323)(233,324)(234,325)
(235,326)(236,327)(237,328)(238,329)(239,316)(240,317)(241,318)(242,319)
(243,320)(244,321)(245,322)(246,330)(247,331)(248,332)(249,333)(250,334)
(251,335)(252,336);;
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,
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,
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)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)
(129,132)(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)
(149,154)(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)
(165,166)(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)(253,295)(254,301)(255,300)(256,299)(257,298)
(258,297)(259,296)(260,302)(261,308)(262,307)(263,306)(264,305)(265,304)
(266,303)(267,309)(268,315)(269,314)(270,313)(271,312)(272,311)(273,310)
(274,316)(275,322)(276,321)(277,320)(278,319)(279,318)(280,317)(281,323)
(282,329)(283,328)(284,327)(285,326)(286,325)(287,324)(288,330)(289,336)
(290,335)(291,334)(292,333)(293,332)(294,331);
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)( 85,275)( 86,274)( 87,280)
( 88,279)( 89,278)( 90,277)( 91,276)( 92,289)( 93,288)( 94,294)( 95,293)
( 96,292)( 97,291)( 98,290)( 99,282)(100,281)(101,287)(102,286)(103,285)
(104,284)(105,283)(106,254)(107,253)(108,259)(109,258)(110,257)(111,256)
(112,255)(113,268)(114,267)(115,273)(116,272)(117,271)(118,270)(119,269)
(120,261)(121,260)(122,266)(123,265)(124,264)(125,263)(126,262)(127,317)
(128,316)(129,322)(130,321)(131,320)(132,319)(133,318)(134,331)(135,330)
(136,336)(137,335)(138,334)(139,333)(140,332)(141,324)(142,323)(143,329)
(144,328)(145,327)(146,326)(147,325)(148,296)(149,295)(150,301)(151,300)
(152,299)(153,298)(154,297)(155,310)(156,309)(157,315)(158,314)(159,313)
(160,312)(161,311)(162,303)(163,302)(164,308)(165,307)(166,306)(167,305)
(168,304);
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, 50)
( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 64, 71)( 65, 72)
( 66, 73)( 67, 74)( 68, 75)( 69, 76)( 70, 77)( 85,113)( 86,114)( 87,115)
( 88,116)( 89,117)( 90,118)( 91,119)( 92,106)( 93,107)( 94,108)( 95,109)
( 96,110)( 97,111)( 98,112)( 99,120)(100,121)(101,122)(102,123)(103,124)
(104,125)(105,126)(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)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)(217,308)
(218,295)(219,296)(220,297)(221,298)(222,299)(223,300)(224,301)(225,309)
(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,323)(233,324)
(234,325)(235,326)(236,327)(237,328)(238,329)(239,316)(240,317)(241,318)
(242,319)(243,320)(244,321)(245,322)(246,330)(247,331)(248,332)(249,333)
(250,334)(251,335)(252,336);
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,
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,
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