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Polytope of Type {3,6,30}

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