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Polytope of Type {30,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6,3}*1080a
if this polytope has a name.
Group : SmallGroup(1080,337)
Rank : 4
Schlafli Type : {30,6,3}
Number of vertices, edges, etc : 30, 90, 9, 3
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   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 : {15,6,3}*540
   3-fold quotients : {30,2,3}*360
   5-fold quotients : {6,6,3}*216a
   6-fold quotients : {15,2,3}*180
   9-fold quotients : {10,2,3}*120
   10-fold quotients : {3,6,3}*108
   15-fold quotients : {6,2,3}*72
   18-fold quotients : {5,2,3}*60
   30-fold quotients : {3,2,3}*36
   45-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 16, 31)
( 17, 33)( 18, 32)( 19, 43)( 20, 45)( 21, 44)( 22, 40)( 23, 42)( 24, 41)
( 25, 37)( 26, 39)( 27, 38)( 28, 34)( 29, 36)( 30, 35)( 47, 48)( 49, 58)
( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 76)( 62, 78)( 63, 77)
( 64, 88)( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70, 82)( 71, 84)
( 72, 83)( 73, 79)( 74, 81)( 75, 80)( 92, 93)( 94,103)( 95,105)( 96,104)
( 97,100)( 98,102)( 99,101)(106,121)(107,123)(108,122)(109,133)(110,135)
(111,134)(112,130)(113,132)(114,131)(115,127)(116,129)(117,128)(118,124)
(119,126)(120,125)(137,138)(139,148)(140,150)(141,149)(142,145)(143,147)
(144,146)(151,166)(152,168)(153,167)(154,178)(155,180)(156,179)(157,175)
(158,177)(159,176)(160,172)(161,174)(162,173)(163,169)(164,171)(165,170)
(182,183)(184,193)(185,195)(186,194)(187,190)(188,192)(189,191)(196,211)
(197,213)(198,212)(199,223)(200,225)(201,224)(202,220)(203,222)(204,221)
(205,217)(206,219)(207,218)(208,214)(209,216)(210,215)(227,228)(229,238)
(230,240)(231,239)(232,235)(233,237)(234,236)(241,256)(242,258)(243,257)
(244,268)(245,270)(246,269)(247,265)(248,267)(249,266)(250,262)(251,264)
(252,263)(253,259)(254,261)(255,260);;
s1 := (  1,154)(  2,156)(  3,155)(  4,151)(  5,153)(  6,152)(  7,163)(  8,165)
(  9,164)( 10,160)( 11,162)( 12,161)( 13,157)( 14,159)( 15,158)( 16,139)
( 17,141)( 18,140)( 19,136)( 20,138)( 21,137)( 22,148)( 23,150)( 24,149)
( 25,145)( 26,147)( 27,146)( 28,142)( 29,144)( 30,143)( 31,169)( 32,171)
( 33,170)( 34,166)( 35,168)( 36,167)( 37,178)( 38,180)( 39,179)( 40,175)
( 41,177)( 42,176)( 43,172)( 44,174)( 45,173)( 46,199)( 47,201)( 48,200)
( 49,196)( 50,198)( 51,197)( 52,208)( 53,210)( 54,209)( 55,205)( 56,207)
( 57,206)( 58,202)( 59,204)( 60,203)( 61,184)( 62,186)( 63,185)( 64,181)
( 65,183)( 66,182)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)( 72,191)
( 73,187)( 74,189)( 75,188)( 76,214)( 77,216)( 78,215)( 79,211)( 80,213)
( 81,212)( 82,223)( 83,225)( 84,224)( 85,220)( 86,222)( 87,221)( 88,217)
( 89,219)( 90,218)( 91,244)( 92,246)( 93,245)( 94,241)( 95,243)( 96,242)
( 97,253)( 98,255)( 99,254)(100,250)(101,252)(102,251)(103,247)(104,249)
(105,248)(106,229)(107,231)(108,230)(109,226)(110,228)(111,227)(112,238)
(113,240)(114,239)(115,235)(116,237)(117,236)(118,232)(119,234)(120,233)
(121,259)(122,261)(123,260)(124,256)(125,258)(126,257)(127,268)(128,270)
(129,269)(130,265)(131,267)(132,266)(133,262)(134,264)(135,263);;
s2 := (  1, 46)(  2, 48)(  3, 47)(  4, 49)(  5, 51)(  6, 50)(  7, 52)(  8, 54)
(  9, 53)( 10, 55)( 11, 57)( 12, 56)( 13, 58)( 14, 60)( 15, 59)( 16, 62)
( 17, 61)( 18, 63)( 19, 65)( 20, 64)( 21, 66)( 22, 68)( 23, 67)( 24, 69)
( 25, 71)( 26, 70)( 27, 72)( 28, 74)( 29, 73)( 30, 75)( 31, 78)( 32, 77)
( 33, 76)( 34, 81)( 35, 80)( 36, 79)( 37, 84)( 38, 83)( 39, 82)( 40, 87)
( 41, 86)( 42, 85)( 43, 90)( 44, 89)( 45, 88)( 92, 93)( 95, 96)( 98, 99)
(101,102)(104,105)(106,107)(109,110)(112,113)(115,116)(118,119)(121,123)
(124,126)(127,129)(130,132)(133,135)(136,181)(137,183)(138,182)(139,184)
(140,186)(141,185)(142,187)(143,189)(144,188)(145,190)(146,192)(147,191)
(148,193)(149,195)(150,194)(151,197)(152,196)(153,198)(154,200)(155,199)
(156,201)(157,203)(158,202)(159,204)(160,206)(161,205)(162,207)(163,209)
(164,208)(165,210)(166,213)(167,212)(168,211)(169,216)(170,215)(171,214)
(172,219)(173,218)(174,217)(175,222)(176,221)(177,220)(178,225)(179,224)
(180,223)(227,228)(230,231)(233,234)(236,237)(239,240)(241,242)(244,245)
(247,248)(250,251)(253,254)(256,258)(259,261)(262,264)(265,267)(268,270);;
s3 := (  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);;
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, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*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*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(270)!(  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)
( 16, 31)( 17, 33)( 18, 32)( 19, 43)( 20, 45)( 21, 44)( 22, 40)( 23, 42)
( 24, 41)( 25, 37)( 26, 39)( 27, 38)( 28, 34)( 29, 36)( 30, 35)( 47, 48)
( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 76)( 62, 78)
( 63, 77)( 64, 88)( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70, 82)
( 71, 84)( 72, 83)( 73, 79)( 74, 81)( 75, 80)( 92, 93)( 94,103)( 95,105)
( 96,104)( 97,100)( 98,102)( 99,101)(106,121)(107,123)(108,122)(109,133)
(110,135)(111,134)(112,130)(113,132)(114,131)(115,127)(116,129)(117,128)
(118,124)(119,126)(120,125)(137,138)(139,148)(140,150)(141,149)(142,145)
(143,147)(144,146)(151,166)(152,168)(153,167)(154,178)(155,180)(156,179)
(157,175)(158,177)(159,176)(160,172)(161,174)(162,173)(163,169)(164,171)
(165,170)(182,183)(184,193)(185,195)(186,194)(187,190)(188,192)(189,191)
(196,211)(197,213)(198,212)(199,223)(200,225)(201,224)(202,220)(203,222)
(204,221)(205,217)(206,219)(207,218)(208,214)(209,216)(210,215)(227,228)
(229,238)(230,240)(231,239)(232,235)(233,237)(234,236)(241,256)(242,258)
(243,257)(244,268)(245,270)(246,269)(247,265)(248,267)(249,266)(250,262)
(251,264)(252,263)(253,259)(254,261)(255,260);
s1 := Sym(270)!(  1,154)(  2,156)(  3,155)(  4,151)(  5,153)(  6,152)(  7,163)
(  8,165)(  9,164)( 10,160)( 11,162)( 12,161)( 13,157)( 14,159)( 15,158)
( 16,139)( 17,141)( 18,140)( 19,136)( 20,138)( 21,137)( 22,148)( 23,150)
( 24,149)( 25,145)( 26,147)( 27,146)( 28,142)( 29,144)( 30,143)( 31,169)
( 32,171)( 33,170)( 34,166)( 35,168)( 36,167)( 37,178)( 38,180)( 39,179)
( 40,175)( 41,177)( 42,176)( 43,172)( 44,174)( 45,173)( 46,199)( 47,201)
( 48,200)( 49,196)( 50,198)( 51,197)( 52,208)( 53,210)( 54,209)( 55,205)
( 56,207)( 57,206)( 58,202)( 59,204)( 60,203)( 61,184)( 62,186)( 63,185)
( 64,181)( 65,183)( 66,182)( 67,193)( 68,195)( 69,194)( 70,190)( 71,192)
( 72,191)( 73,187)( 74,189)( 75,188)( 76,214)( 77,216)( 78,215)( 79,211)
( 80,213)( 81,212)( 82,223)( 83,225)( 84,224)( 85,220)( 86,222)( 87,221)
( 88,217)( 89,219)( 90,218)( 91,244)( 92,246)( 93,245)( 94,241)( 95,243)
( 96,242)( 97,253)( 98,255)( 99,254)(100,250)(101,252)(102,251)(103,247)
(104,249)(105,248)(106,229)(107,231)(108,230)(109,226)(110,228)(111,227)
(112,238)(113,240)(114,239)(115,235)(116,237)(117,236)(118,232)(119,234)
(120,233)(121,259)(122,261)(123,260)(124,256)(125,258)(126,257)(127,268)
(128,270)(129,269)(130,265)(131,267)(132,266)(133,262)(134,264)(135,263);
s2 := Sym(270)!(  1, 46)(  2, 48)(  3, 47)(  4, 49)(  5, 51)(  6, 50)(  7, 52)
(  8, 54)(  9, 53)( 10, 55)( 11, 57)( 12, 56)( 13, 58)( 14, 60)( 15, 59)
( 16, 62)( 17, 61)( 18, 63)( 19, 65)( 20, 64)( 21, 66)( 22, 68)( 23, 67)
( 24, 69)( 25, 71)( 26, 70)( 27, 72)( 28, 74)( 29, 73)( 30, 75)( 31, 78)
( 32, 77)( 33, 76)( 34, 81)( 35, 80)( 36, 79)( 37, 84)( 38, 83)( 39, 82)
( 40, 87)( 41, 86)( 42, 85)( 43, 90)( 44, 89)( 45, 88)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(106,107)(109,110)(112,113)(115,116)(118,119)
(121,123)(124,126)(127,129)(130,132)(133,135)(136,181)(137,183)(138,182)
(139,184)(140,186)(141,185)(142,187)(143,189)(144,188)(145,190)(146,192)
(147,191)(148,193)(149,195)(150,194)(151,197)(152,196)(153,198)(154,200)
(155,199)(156,201)(157,203)(158,202)(159,204)(160,206)(161,205)(162,207)
(163,209)(164,208)(165,210)(166,213)(167,212)(168,211)(169,216)(170,215)
(171,214)(172,219)(173,218)(174,217)(175,222)(176,221)(177,220)(178,225)
(179,224)(180,223)(227,228)(230,231)(233,234)(236,237)(239,240)(241,242)
(244,245)(247,248)(250,251)(253,254)(256,258)(259,261)(262,264)(265,267)
(268,270);
s3 := 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);
poly := sub<Sym(270)|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, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*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*s0*s1*s0*s1 >; 
 
References : None.
to this polytope