Polytope of Type {6,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,28}*1344e
if this polytope has a name.
Group : SmallGroup(1344,11328)
Rank : 3
Schlafli Type : {6,28}
Number of vertices, edges, etc : 24, 336, 112
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 : {6,28}*336a, {6,28}*336b
   7-fold quotients : {6,4}*192b
   8-fold quotients : {6,14}*168
   12-fold quotients : {2,28}*112
   14-fold quotients : {6,4}*96
   24-fold quotients : {2,14}*56
   28-fold quotients : {6,4}*48a, {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 : {2,4}*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)(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);;
s1 := (  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);;
s2 := (  1,174)(  2,173)(  3,176)(  4,175)(  5,170)(  6,169)(  7,172)(  8,171)
(  9,194)( 10,193)( 11,196)( 12,195)( 13,190)( 14,189)( 15,192)( 16,191)
( 17,186)( 18,185)( 19,188)( 20,187)( 21,182)( 22,181)( 23,184)( 24,183)
( 25,178)( 26,177)( 27,180)( 28,179)( 29,202)( 30,201)( 31,204)( 32,203)
( 33,198)( 34,197)( 35,200)( 36,199)( 37,222)( 38,221)( 39,224)( 40,223)
( 41,218)( 42,217)( 43,220)( 44,219)( 45,214)( 46,213)( 47,216)( 48,215)
( 49,210)( 50,209)( 51,212)( 52,211)( 53,206)( 54,205)( 55,208)( 56,207)
( 57,230)( 58,229)( 59,232)( 60,231)( 61,226)( 62,225)( 63,228)( 64,227)
( 65,250)( 66,249)( 67,252)( 68,251)( 69,246)( 70,245)( 71,248)( 72,247)
( 73,242)( 74,241)( 75,244)( 76,243)( 77,238)( 78,237)( 79,240)( 80,239)
( 81,234)( 82,233)( 83,236)( 84,235)( 85,258)( 86,257)( 87,260)( 88,259)
( 89,254)( 90,253)( 91,256)( 92,255)( 93,278)( 94,277)( 95,280)( 96,279)
( 97,274)( 98,273)( 99,276)(100,275)(101,270)(102,269)(103,272)(104,271)
(105,266)(106,265)(107,268)(108,267)(109,262)(110,261)(111,264)(112,263)
(113,286)(114,285)(115,288)(116,287)(117,282)(118,281)(119,284)(120,283)
(121,306)(122,305)(123,308)(124,307)(125,302)(126,301)(127,304)(128,303)
(129,298)(130,297)(131,300)(132,299)(133,294)(134,293)(135,296)(136,295)
(137,290)(138,289)(139,292)(140,291)(141,314)(142,313)(143,316)(144,315)
(145,310)(146,309)(147,312)(148,311)(149,334)(150,333)(151,336)(152,335)
(153,330)(154,329)(155,332)(156,331)(157,326)(158,325)(159,328)(160,327)
(161,322)(162,321)(163,324)(164,323)(165,318)(166,317)(167,320)(168,319);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*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*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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)(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);
s1 := 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);
s2 := Sym(336)!(  1,174)(  2,173)(  3,176)(  4,175)(  5,170)(  6,169)(  7,172)
(  8,171)(  9,194)( 10,193)( 11,196)( 12,195)( 13,190)( 14,189)( 15,192)
( 16,191)( 17,186)( 18,185)( 19,188)( 20,187)( 21,182)( 22,181)( 23,184)
( 24,183)( 25,178)( 26,177)( 27,180)( 28,179)( 29,202)( 30,201)( 31,204)
( 32,203)( 33,198)( 34,197)( 35,200)( 36,199)( 37,222)( 38,221)( 39,224)
( 40,223)( 41,218)( 42,217)( 43,220)( 44,219)( 45,214)( 46,213)( 47,216)
( 48,215)( 49,210)( 50,209)( 51,212)( 52,211)( 53,206)( 54,205)( 55,208)
( 56,207)( 57,230)( 58,229)( 59,232)( 60,231)( 61,226)( 62,225)( 63,228)
( 64,227)( 65,250)( 66,249)( 67,252)( 68,251)( 69,246)( 70,245)( 71,248)
( 72,247)( 73,242)( 74,241)( 75,244)( 76,243)( 77,238)( 78,237)( 79,240)
( 80,239)( 81,234)( 82,233)( 83,236)( 84,235)( 85,258)( 86,257)( 87,260)
( 88,259)( 89,254)( 90,253)( 91,256)( 92,255)( 93,278)( 94,277)( 95,280)
( 96,279)( 97,274)( 98,273)( 99,276)(100,275)(101,270)(102,269)(103,272)
(104,271)(105,266)(106,265)(107,268)(108,267)(109,262)(110,261)(111,264)
(112,263)(113,286)(114,285)(115,288)(116,287)(117,282)(118,281)(119,284)
(120,283)(121,306)(122,305)(123,308)(124,307)(125,302)(126,301)(127,304)
(128,303)(129,298)(130,297)(131,300)(132,299)(133,294)(134,293)(135,296)
(136,295)(137,290)(138,289)(139,292)(140,291)(141,314)(142,313)(143,316)
(144,315)(145,310)(146,309)(147,312)(148,311)(149,334)(150,333)(151,336)
(152,335)(153,330)(154,329)(155,332)(156,331)(157,326)(158,325)(159,328)
(160,327)(161,322)(162,321)(163,324)(164,323)(165,318)(166,317)(167,320)
(168,319);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*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*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope