Polytope of Type {6,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30}*1080b
if this polytope has a name.
Group : SmallGroup(1080,337)
Rank : 3
Schlafli Type : {6,30}
Number of vertices, edges, etc : 18, 270, 90
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 6
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 : {6,15}*540
   3-fold quotients : {6,30}*360c
   5-fold quotients : {6,6}*216a
   6-fold quotients : {6,15}*180
   9-fold quotients : {2,30}*120
   10-fold quotients : {6,3}*108
   15-fold quotients : {6,6}*72b
   18-fold quotients : {2,15}*60
   27-fold quotients : {2,10}*40
   30-fold quotients : {6,3}*36
   45-fold quotients : {2,6}*24
   54-fold quotients : {2,5}*20
   90-fold quotients : {2,3}*12
   135-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope