Polytope of Type {4,6,28}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,28}*1344b
if this polytope has a name.
Group : SmallGroup(1344,11328)
Rank : 4
Schlafli Type : {4,6,28}
Number of vertices, edges, etc : 4, 12, 84, 28
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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 : {4,6,14}*672b
7-fold quotients : {4,6,4}*192c
14-fold quotients : {4,6,2}*96c
28-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336);;
s1 := ( 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)(171,172)(175,176)
(179,180)(183,184)(187,188)(191,192)(195,196)(197,225)(198,226)(199,228)
(200,227)(201,229)(202,230)(203,232)(204,231)(205,233)(206,234)(207,236)
(208,235)(209,237)(210,238)(211,240)(212,239)(213,241)(214,242)(215,244)
(216,243)(217,245)(218,246)(219,248)(220,247)(221,249)(222,250)(223,252)
(224,251)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)(279,280)
(281,309)(282,310)(283,312)(284,311)(285,313)(286,314)(287,316)(288,315)
(289,317)(290,318)(291,320)(292,319)(293,321)(294,322)(295,324)(296,323)
(297,325)(298,326)(299,328)(300,327)(301,329)(302,330)(303,332)(304,331)
(305,333)(306,334)(307,336)(308,335);;
s2 := ( 1, 29)( 2, 32)( 3, 31)( 4, 30)( 5, 53)( 6, 56)( 7, 55)( 8, 54)
( 9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 45)( 14, 48)( 15, 47)( 16, 46)
( 17, 41)( 18, 44)( 19, 43)( 20, 42)( 21, 37)( 22, 40)( 23, 39)( 24, 38)
( 25, 33)( 26, 36)( 27, 35)( 28, 34)( 58, 60)( 61, 81)( 62, 84)( 63, 83)
( 64, 82)( 65, 77)( 66, 80)( 67, 79)( 68, 78)( 69, 73)( 70, 76)( 71, 75)
( 72, 74)( 85,113)( 86,116)( 87,115)( 88,114)( 89,137)( 90,140)( 91,139)
( 92,138)( 93,133)( 94,136)( 95,135)( 96,134)( 97,129)( 98,132)( 99,131)
(100,130)(101,125)(102,128)(103,127)(104,126)(105,121)(106,124)(107,123)
(108,122)(109,117)(110,120)(111,119)(112,118)(142,144)(145,165)(146,168)
(147,167)(148,166)(149,161)(150,164)(151,163)(152,162)(153,157)(154,160)
(155,159)(156,158)(169,281)(170,284)(171,283)(172,282)(173,305)(174,308)
(175,307)(176,306)(177,301)(178,304)(179,303)(180,302)(181,297)(182,300)
(183,299)(184,298)(185,293)(186,296)(187,295)(188,294)(189,289)(190,292)
(191,291)(192,290)(193,285)(194,288)(195,287)(196,286)(197,253)(198,256)
(199,255)(200,254)(201,277)(202,280)(203,279)(204,278)(205,273)(206,276)
(207,275)(208,274)(209,269)(210,272)(211,271)(212,270)(213,265)(214,268)
(215,267)(216,266)(217,261)(218,264)(219,263)(220,262)(221,257)(222,260)
(223,259)(224,258)(225,309)(226,312)(227,311)(228,310)(229,333)(230,336)
(231,335)(232,334)(233,329)(234,332)(235,331)(236,330)(237,325)(238,328)
(239,327)(240,326)(241,321)(242,324)(243,323)(244,322)(245,317)(246,320)
(247,319)(248,318)(249,313)(250,316)(251,315)(252,314);;
s3 := ( 1,173)( 2,174)( 3,175)( 4,176)( 5,169)( 6,170)( 7,171)( 8,172)
( 9,193)( 10,194)( 11,195)( 12,196)( 13,189)( 14,190)( 15,191)( 16,192)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,181)( 22,182)( 23,183)( 24,184)
( 25,177)( 26,178)( 27,179)( 28,180)( 29,201)( 30,202)( 31,203)( 32,204)
( 33,197)( 34,198)( 35,199)( 36,200)( 37,221)( 38,222)( 39,223)( 40,224)
( 41,217)( 42,218)( 43,219)( 44,220)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,209)( 50,210)( 51,211)( 52,212)( 53,205)( 54,206)( 55,207)( 56,208)
( 57,229)( 58,230)( 59,231)( 60,232)( 61,225)( 62,226)( 63,227)( 64,228)
( 65,249)( 66,250)( 67,251)( 68,252)( 69,245)( 70,246)( 71,247)( 72,248)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,237)( 78,238)( 79,239)( 80,240)
( 81,233)( 82,234)( 83,235)( 84,236)( 85,257)( 86,258)( 87,259)( 88,260)
( 89,253)( 90,254)( 91,255)( 92,256)( 93,277)( 94,278)( 95,279)( 96,280)
( 97,273)( 98,274)( 99,275)(100,276)(101,269)(102,270)(103,271)(104,272)
(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)(112,264)
(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)(120,284)
(121,305)(122,306)(123,307)(124,308)(125,301)(126,302)(127,303)(128,304)
(129,297)(130,298)(131,299)(132,300)(133,293)(134,294)(135,295)(136,296)
(137,289)(138,290)(139,291)(140,292)(141,313)(142,314)(143,315)(144,316)
(145,309)(146,310)(147,311)(148,312)(149,333)(150,334)(151,335)(152,336)
(153,329)(154,330)(155,331)(156,332)(157,325)(158,326)(159,327)(160,328)
(161,321)(162,322)(163,323)(164,324)(165,317)(166,318)(167,319)(168,320);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(336)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336);
s1 := 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)(171,172)
(175,176)(179,180)(183,184)(187,188)(191,192)(195,196)(197,225)(198,226)
(199,228)(200,227)(201,229)(202,230)(203,232)(204,231)(205,233)(206,234)
(207,236)(208,235)(209,237)(210,238)(211,240)(212,239)(213,241)(214,242)
(215,244)(216,243)(217,245)(218,246)(219,248)(220,247)(221,249)(222,250)
(223,252)(224,251)(255,256)(259,260)(263,264)(267,268)(271,272)(275,276)
(279,280)(281,309)(282,310)(283,312)(284,311)(285,313)(286,314)(287,316)
(288,315)(289,317)(290,318)(291,320)(292,319)(293,321)(294,322)(295,324)
(296,323)(297,325)(298,326)(299,328)(300,327)(301,329)(302,330)(303,332)
(304,331)(305,333)(306,334)(307,336)(308,335);
s2 := Sym(336)!( 1, 29)( 2, 32)( 3, 31)( 4, 30)( 5, 53)( 6, 56)( 7, 55)
( 8, 54)( 9, 49)( 10, 52)( 11, 51)( 12, 50)( 13, 45)( 14, 48)( 15, 47)
( 16, 46)( 17, 41)( 18, 44)( 19, 43)( 20, 42)( 21, 37)( 22, 40)( 23, 39)
( 24, 38)( 25, 33)( 26, 36)( 27, 35)( 28, 34)( 58, 60)( 61, 81)( 62, 84)
( 63, 83)( 64, 82)( 65, 77)( 66, 80)( 67, 79)( 68, 78)( 69, 73)( 70, 76)
( 71, 75)( 72, 74)( 85,113)( 86,116)( 87,115)( 88,114)( 89,137)( 90,140)
( 91,139)( 92,138)( 93,133)( 94,136)( 95,135)( 96,134)( 97,129)( 98,132)
( 99,131)(100,130)(101,125)(102,128)(103,127)(104,126)(105,121)(106,124)
(107,123)(108,122)(109,117)(110,120)(111,119)(112,118)(142,144)(145,165)
(146,168)(147,167)(148,166)(149,161)(150,164)(151,163)(152,162)(153,157)
(154,160)(155,159)(156,158)(169,281)(170,284)(171,283)(172,282)(173,305)
(174,308)(175,307)(176,306)(177,301)(178,304)(179,303)(180,302)(181,297)
(182,300)(183,299)(184,298)(185,293)(186,296)(187,295)(188,294)(189,289)
(190,292)(191,291)(192,290)(193,285)(194,288)(195,287)(196,286)(197,253)
(198,256)(199,255)(200,254)(201,277)(202,280)(203,279)(204,278)(205,273)
(206,276)(207,275)(208,274)(209,269)(210,272)(211,271)(212,270)(213,265)
(214,268)(215,267)(216,266)(217,261)(218,264)(219,263)(220,262)(221,257)
(222,260)(223,259)(224,258)(225,309)(226,312)(227,311)(228,310)(229,333)
(230,336)(231,335)(232,334)(233,329)(234,332)(235,331)(236,330)(237,325)
(238,328)(239,327)(240,326)(241,321)(242,324)(243,323)(244,322)(245,317)
(246,320)(247,319)(248,318)(249,313)(250,316)(251,315)(252,314);
s3 := Sym(336)!( 1,173)( 2,174)( 3,175)( 4,176)( 5,169)( 6,170)( 7,171)
( 8,172)( 9,193)( 10,194)( 11,195)( 12,196)( 13,189)( 14,190)( 15,191)
( 16,192)( 17,185)( 18,186)( 19,187)( 20,188)( 21,181)( 22,182)( 23,183)
( 24,184)( 25,177)( 26,178)( 27,179)( 28,180)( 29,201)( 30,202)( 31,203)
( 32,204)( 33,197)( 34,198)( 35,199)( 36,200)( 37,221)( 38,222)( 39,223)
( 40,224)( 41,217)( 42,218)( 43,219)( 44,220)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,209)( 50,210)( 51,211)( 52,212)( 53,205)( 54,206)( 55,207)
( 56,208)( 57,229)( 58,230)( 59,231)( 60,232)( 61,225)( 62,226)( 63,227)
( 64,228)( 65,249)( 66,250)( 67,251)( 68,252)( 69,245)( 70,246)( 71,247)
( 72,248)( 73,241)( 74,242)( 75,243)( 76,244)( 77,237)( 78,238)( 79,239)
( 80,240)( 81,233)( 82,234)( 83,235)( 84,236)( 85,257)( 86,258)( 87,259)
( 88,260)( 89,253)( 90,254)( 91,255)( 92,256)( 93,277)( 94,278)( 95,279)
( 96,280)( 97,273)( 98,274)( 99,275)(100,276)(101,269)(102,270)(103,271)
(104,272)(105,265)(106,266)(107,267)(108,268)(109,261)(110,262)(111,263)
(112,264)(113,285)(114,286)(115,287)(116,288)(117,281)(118,282)(119,283)
(120,284)(121,305)(122,306)(123,307)(124,308)(125,301)(126,302)(127,303)
(128,304)(129,297)(130,298)(131,299)(132,300)(133,293)(134,294)(135,295)
(136,296)(137,289)(138,290)(139,291)(140,292)(141,313)(142,314)(143,315)
(144,316)(145,309)(146,310)(147,311)(148,312)(149,333)(150,334)(151,335)
(152,336)(153,329)(154,330)(155,331)(156,332)(157,325)(158,326)(159,327)
(160,328)(161,321)(162,322)(163,323)(164,324)(165,317)(166,318)(167,319)
(168,320);
poly := sub<Sym(336)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope