Polytope of Type {6,30,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,30,4}*1920
if this polytope has a name.
Group : SmallGroup(1920,240151)
Rank : 4
Schlafli Type : {6,30,4}
Number of vertices, edges, etc : 8, 120, 80, 4
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
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 : {6,30,2}*960
4-fold quotients : {6,15,2}*480
5-fold quotients : {6,6,4}*384
10-fold quotients : {3,6,4}*192, {6,6,2}*192
12-fold quotients : {2,10,4}*160
20-fold quotients : {3,6,2}*96, {6,3,2}*96
24-fold quotients : {2,10,2}*80
40-fold quotients : {3,3,2}*48
48-fold quotients : {2,5,2}*40
60-fold quotients : {2,2,4}*32
120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)( 23, 44)
( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)( 31, 52)
( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)( 39, 60)
( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)( 82,102)
( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)( 98,118)
( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)(141,161)
(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)(149,169)
(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)(157,177)
(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)(199,200)
(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)(208,227)
(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)(216,235)
(217,237)(218,238)(219,240)(220,239);;
s1 := ( 1, 21)( 2, 24)( 3, 23)( 4, 22)( 5, 37)( 6, 40)( 7, 39)( 8, 38)
( 9, 33)( 10, 36)( 11, 35)( 12, 34)( 13, 29)( 14, 32)( 15, 31)( 16, 30)
( 17, 25)( 18, 28)( 19, 27)( 20, 26)( 42, 44)( 45, 57)( 46, 60)( 47, 59)
( 48, 58)( 49, 53)( 50, 56)( 51, 55)( 52, 54)( 61, 81)( 62, 84)( 63, 83)
( 64, 82)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)( 71, 95)
( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)( 79, 87)
( 80, 86)(102,104)(105,117)(106,120)(107,119)(108,118)(109,113)(110,116)
(111,115)(112,114)(121,141)(122,144)(123,143)(124,142)(125,157)(126,160)
(127,159)(128,158)(129,153)(130,156)(131,155)(132,154)(133,149)(134,152)
(135,151)(136,150)(137,145)(138,148)(139,147)(140,146)(162,164)(165,177)
(166,180)(167,179)(168,178)(169,173)(170,176)(171,175)(172,174)(181,201)
(182,204)(183,203)(184,202)(185,217)(186,220)(187,219)(188,218)(189,213)
(190,216)(191,215)(192,214)(193,209)(194,212)(195,211)(196,210)(197,205)
(198,208)(199,207)(200,206)(222,224)(225,237)(226,240)(227,239)(228,238)
(229,233)(230,236)(231,235)(232,234);;
s2 := ( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9, 18)( 10, 17)( 11, 19)( 12, 20)
( 13, 14)( 21, 46)( 22, 45)( 23, 47)( 24, 48)( 25, 42)( 26, 41)( 27, 43)
( 28, 44)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 54)( 34, 53)( 35, 55)
( 36, 56)( 37, 50)( 38, 49)( 39, 51)( 40, 52)( 61, 66)( 62, 65)( 63, 67)
( 64, 68)( 69, 78)( 70, 77)( 71, 79)( 72, 80)( 73, 74)( 81,106)( 82,105)
( 83,107)( 84,108)( 85,102)( 86,101)( 87,103)( 88,104)( 89,118)( 90,117)
( 91,119)( 92,120)( 93,114)( 94,113)( 95,115)( 96,116)( 97,110)( 98,109)
( 99,111)(100,112)(121,186)(122,185)(123,187)(124,188)(125,182)(126,181)
(127,183)(128,184)(129,198)(130,197)(131,199)(132,200)(133,194)(134,193)
(135,195)(136,196)(137,190)(138,189)(139,191)(140,192)(141,226)(142,225)
(143,227)(144,228)(145,222)(146,221)(147,223)(148,224)(149,238)(150,237)
(151,239)(152,240)(153,234)(154,233)(155,235)(156,236)(157,230)(158,229)
(159,231)(160,232)(161,206)(162,205)(163,207)(164,208)(165,202)(166,201)
(167,203)(168,204)(169,218)(170,217)(171,219)(172,220)(173,214)(174,213)
(175,215)(176,216)(177,210)(178,209)(179,211)(180,212);;
s3 := ( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)
( 9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)
( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)
( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)
( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)( 40,160)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)
( 81,201)( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)( 88,208)
( 89,209)( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)( 96,216)
( 97,217)( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)(104,224)
(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)
(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(240)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)
( 23, 44)( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)
( 31, 52)( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)
( 39, 60)( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)
( 82,102)( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)
( 90,110)( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)
( 98,118)( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)
(141,161)(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)
(149,169)(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)
(157,177)(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)
(199,200)(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)
(208,227)(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)
(216,235)(217,237)(218,238)(219,240)(220,239);
s1 := Sym(240)!( 1, 21)( 2, 24)( 3, 23)( 4, 22)( 5, 37)( 6, 40)( 7, 39)
( 8, 38)( 9, 33)( 10, 36)( 11, 35)( 12, 34)( 13, 29)( 14, 32)( 15, 31)
( 16, 30)( 17, 25)( 18, 28)( 19, 27)( 20, 26)( 42, 44)( 45, 57)( 46, 60)
( 47, 59)( 48, 58)( 49, 53)( 50, 56)( 51, 55)( 52, 54)( 61, 81)( 62, 84)
( 63, 83)( 64, 82)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69, 93)( 70, 96)
( 71, 95)( 72, 94)( 73, 89)( 74, 92)( 75, 91)( 76, 90)( 77, 85)( 78, 88)
( 79, 87)( 80, 86)(102,104)(105,117)(106,120)(107,119)(108,118)(109,113)
(110,116)(111,115)(112,114)(121,141)(122,144)(123,143)(124,142)(125,157)
(126,160)(127,159)(128,158)(129,153)(130,156)(131,155)(132,154)(133,149)
(134,152)(135,151)(136,150)(137,145)(138,148)(139,147)(140,146)(162,164)
(165,177)(166,180)(167,179)(168,178)(169,173)(170,176)(171,175)(172,174)
(181,201)(182,204)(183,203)(184,202)(185,217)(186,220)(187,219)(188,218)
(189,213)(190,216)(191,215)(192,214)(193,209)(194,212)(195,211)(196,210)
(197,205)(198,208)(199,207)(200,206)(222,224)(225,237)(226,240)(227,239)
(228,238)(229,233)(230,236)(231,235)(232,234);
s2 := Sym(240)!( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9, 18)( 10, 17)( 11, 19)
( 12, 20)( 13, 14)( 21, 46)( 22, 45)( 23, 47)( 24, 48)( 25, 42)( 26, 41)
( 27, 43)( 28, 44)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 54)( 34, 53)
( 35, 55)( 36, 56)( 37, 50)( 38, 49)( 39, 51)( 40, 52)( 61, 66)( 62, 65)
( 63, 67)( 64, 68)( 69, 78)( 70, 77)( 71, 79)( 72, 80)( 73, 74)( 81,106)
( 82,105)( 83,107)( 84,108)( 85,102)( 86,101)( 87,103)( 88,104)( 89,118)
( 90,117)( 91,119)( 92,120)( 93,114)( 94,113)( 95,115)( 96,116)( 97,110)
( 98,109)( 99,111)(100,112)(121,186)(122,185)(123,187)(124,188)(125,182)
(126,181)(127,183)(128,184)(129,198)(130,197)(131,199)(132,200)(133,194)
(134,193)(135,195)(136,196)(137,190)(138,189)(139,191)(140,192)(141,226)
(142,225)(143,227)(144,228)(145,222)(146,221)(147,223)(148,224)(149,238)
(150,237)(151,239)(152,240)(153,234)(154,233)(155,235)(156,236)(157,230)
(158,229)(159,231)(160,232)(161,206)(162,205)(163,207)(164,208)(165,202)
(166,201)(167,203)(168,204)(169,218)(170,217)(171,219)(172,220)(173,214)
(174,213)(175,215)(176,216)(177,210)(178,209)(179,211)(180,212);
s3 := Sym(240)!( 1,121)( 2,122)( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)
( 8,128)( 9,129)( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)
( 16,136)( 17,137)( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)
( 24,144)( 25,145)( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)
( 32,152)( 33,153)( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)
( 40,160)( 41,161)( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)
( 48,168)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,193)( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)
( 80,200)( 81,201)( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)
( 88,208)( 89,209)( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)
( 96,216)( 97,217)( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)
(104,224)(105,225)(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)
(112,232)(113,233)(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)
(120,240);
poly := sub<Sym(240)|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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >;
References : None.
to this polytope