Polytope of Type {10,55}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,55}*1100
if this polytope has a name.
Group : SmallGroup(1100,44)
Rank : 3
Schlafli Type : {10,55}
Number of vertices, edges, etc : 10, 275, 55
Order of s₀s₁s₂ : 110
Order of s₀s₁s₂s₁ : 10
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) :
   5-fold quotients : {2,55}*220
   11-fold quotients : {10,5}*100
   25-fold quotients : {2,11}*44
   55-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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