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.
Irregular Quotients (of which this is a minimal cover):
None.
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, s1*s2*s3*s1*s2*s1*s2*s3*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,
s1*s2*s3*s1*s2*s1*s2*s3*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