Polytope of Type {4,30,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,30,6}*1920
if this polytope has a name.
Group : SmallGroup(1920,240151)
Rank : 4
Schlafli Type : {4,30,6}
Number of vertices, edges, etc : 4, 80, 120, 8
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 : {2,30,6}*960
   4-fold quotients : {2,15,6}*480
   5-fold quotients : {4,6,6}*384
   10-fold quotients : {4,6,3}*192, {2,6,6}*192
   12-fold quotients : {4,10,2}*160
   20-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,10,2}*80
   40-fold quotients : {2,3,3}*48
   48-fold quotients : {2,5,2}*40
   60-fold quotients : {4,2,2}*32
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)
(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)
(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)(144,204)
(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)(152,212)
(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)(160,220)
(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)(168,228)
(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)
(177,237)(178,238)(179,239)(180,240);;
s1 := (  1,121)(  2,122)(  3,124)(  4,123)(  5,137)(  6,138)(  7,140)(  8,139)
(  9,133)( 10,134)( 11,136)( 12,135)( 13,129)( 14,130)( 15,132)( 16,131)
( 17,125)( 18,126)( 19,128)( 20,127)( 21,161)( 22,162)( 23,164)( 24,163)
( 25,177)( 26,178)( 27,180)( 28,179)( 29,173)( 30,174)( 31,176)( 32,175)
( 33,169)( 34,170)( 35,172)( 36,171)( 37,165)( 38,166)( 39,168)( 40,167)
( 41,141)( 42,142)( 43,144)( 44,143)( 45,157)( 46,158)( 47,160)( 48,159)
( 49,153)( 50,154)( 51,156)( 52,155)( 53,149)( 54,150)( 55,152)( 56,151)
( 57,145)( 58,146)( 59,148)( 60,147)( 61,181)( 62,182)( 63,184)( 64,183)
( 65,197)( 66,198)( 67,200)( 68,199)( 69,193)( 70,194)( 71,196)( 72,195)
( 73,189)( 74,190)( 75,192)( 76,191)( 77,185)( 78,186)( 79,188)( 80,187)
( 81,221)( 82,222)( 83,224)( 84,223)( 85,237)( 86,238)( 87,240)( 88,239)
( 89,233)( 90,234)( 91,236)( 92,235)( 93,229)( 94,230)( 95,232)( 96,231)
( 97,225)( 98,226)( 99,228)(100,227)(101,201)(102,202)(103,204)(104,203)
(105,217)(106,218)(107,220)(108,219)(109,213)(110,214)(111,216)(112,215)
(113,209)(114,210)(115,212)(116,211)(117,205)(118,206)(119,208)(120,207);;
s2 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 21)(  6, 24)(  7, 23)(  8, 22)
(  9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 33)( 14, 36)( 15, 35)( 16, 34)
( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 41, 45)( 42, 48)( 43, 47)( 44, 46)
( 49, 57)( 50, 60)( 51, 59)( 52, 58)( 54, 56)( 61, 85)( 62, 88)( 63, 87)
( 64, 86)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 97)( 70,100)( 71, 99)
( 72, 98)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 89)( 78, 92)( 79, 91)
( 80, 90)(101,105)(102,108)(103,107)(104,106)(109,117)(110,120)(111,119)
(112,118)(114,116)(121,145)(122,148)(123,147)(124,146)(125,141)(126,144)
(127,143)(128,142)(129,157)(130,160)(131,159)(132,158)(133,153)(134,156)
(135,155)(136,154)(137,149)(138,152)(139,151)(140,150)(161,165)(162,168)
(163,167)(164,166)(169,177)(170,180)(171,179)(172,178)(174,176)(181,205)
(182,208)(183,207)(184,206)(185,201)(186,204)(187,203)(188,202)(189,217)
(190,220)(191,219)(192,218)(193,213)(194,216)(195,215)(196,214)(197,209)
(198,212)(199,211)(200,210)(221,225)(222,228)(223,227)(224,226)(229,237)
(230,240)(231,239)(232,238)(234,236);;
s3 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 42)( 22, 41)( 23, 43)
( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 50)( 30, 49)( 31, 51)
( 32, 52)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 39, 59)
( 40, 60)( 61, 62)( 65, 66)( 69, 70)( 73, 74)( 77, 78)( 81,102)( 82,101)
( 83,103)( 84,104)( 85,106)( 86,105)( 87,107)( 88,108)( 89,110)( 90,109)
( 91,111)( 92,112)( 93,114)( 94,113)( 95,115)( 96,116)( 97,118)( 98,117)
( 99,119)(100,120)(121,122)(125,126)(129,130)(133,134)(137,138)(141,162)
(142,161)(143,163)(144,164)(145,166)(146,165)(147,167)(148,168)(149,170)
(150,169)(151,171)(152,172)(153,174)(154,173)(155,175)(156,176)(157,178)
(158,177)(159,179)(160,180)(181,182)(185,186)(189,190)(193,194)(197,198)
(201,222)(202,221)(203,223)(204,224)(205,226)(206,225)(207,227)(208,228)
(209,230)(210,229)(211,231)(212,232)(213,234)(214,233)(215,235)(216,236)
(217,238)(218,237)(219,239)(220,240);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(240)!(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)
(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)
(136,196)(137,197)(138,198)(139,199)(140,200)(141,201)(142,202)(143,203)
(144,204)(145,205)(146,206)(147,207)(148,208)(149,209)(150,210)(151,211)
(152,212)(153,213)(154,214)(155,215)(156,216)(157,217)(158,218)(159,219)
(160,220)(161,221)(162,222)(163,223)(164,224)(165,225)(166,226)(167,227)
(168,228)(169,229)(170,230)(171,231)(172,232)(173,233)(174,234)(175,235)
(176,236)(177,237)(178,238)(179,239)(180,240);
s1 := Sym(240)!(  1,121)(  2,122)(  3,124)(  4,123)(  5,137)(  6,138)(  7,140)
(  8,139)(  9,133)( 10,134)( 11,136)( 12,135)( 13,129)( 14,130)( 15,132)
( 16,131)( 17,125)( 18,126)( 19,128)( 20,127)( 21,161)( 22,162)( 23,164)
( 24,163)( 25,177)( 26,178)( 27,180)( 28,179)( 29,173)( 30,174)( 31,176)
( 32,175)( 33,169)( 34,170)( 35,172)( 36,171)( 37,165)( 38,166)( 39,168)
( 40,167)( 41,141)( 42,142)( 43,144)( 44,143)( 45,157)( 46,158)( 47,160)
( 48,159)( 49,153)( 50,154)( 51,156)( 52,155)( 53,149)( 54,150)( 55,152)
( 56,151)( 57,145)( 58,146)( 59,148)( 60,147)( 61,181)( 62,182)( 63,184)
( 64,183)( 65,197)( 66,198)( 67,200)( 68,199)( 69,193)( 70,194)( 71,196)
( 72,195)( 73,189)( 74,190)( 75,192)( 76,191)( 77,185)( 78,186)( 79,188)
( 80,187)( 81,221)( 82,222)( 83,224)( 84,223)( 85,237)( 86,238)( 87,240)
( 88,239)( 89,233)( 90,234)( 91,236)( 92,235)( 93,229)( 94,230)( 95,232)
( 96,231)( 97,225)( 98,226)( 99,228)(100,227)(101,201)(102,202)(103,204)
(104,203)(105,217)(106,218)(107,220)(108,219)(109,213)(110,214)(111,216)
(112,215)(113,209)(114,210)(115,212)(116,211)(117,205)(118,206)(119,208)
(120,207);
s2 := Sym(240)!(  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 21)(  6, 24)(  7, 23)
(  8, 22)(  9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 41, 45)( 42, 48)( 43, 47)
( 44, 46)( 49, 57)( 50, 60)( 51, 59)( 52, 58)( 54, 56)( 61, 85)( 62, 88)
( 63, 87)( 64, 86)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 97)( 70,100)
( 71, 99)( 72, 98)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 89)( 78, 92)
( 79, 91)( 80, 90)(101,105)(102,108)(103,107)(104,106)(109,117)(110,120)
(111,119)(112,118)(114,116)(121,145)(122,148)(123,147)(124,146)(125,141)
(126,144)(127,143)(128,142)(129,157)(130,160)(131,159)(132,158)(133,153)
(134,156)(135,155)(136,154)(137,149)(138,152)(139,151)(140,150)(161,165)
(162,168)(163,167)(164,166)(169,177)(170,180)(171,179)(172,178)(174,176)
(181,205)(182,208)(183,207)(184,206)(185,201)(186,204)(187,203)(188,202)
(189,217)(190,220)(191,219)(192,218)(193,213)(194,216)(195,215)(196,214)
(197,209)(198,212)(199,211)(200,210)(221,225)(222,228)(223,227)(224,226)
(229,237)(230,240)(231,239)(232,238)(234,236);
s3 := Sym(240)!(  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 42)( 22, 41)
( 23, 43)( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 50)( 30, 49)
( 31, 51)( 32, 52)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 37, 58)( 38, 57)
( 39, 59)( 40, 60)( 61, 62)( 65, 66)( 69, 70)( 73, 74)( 77, 78)( 81,102)
( 82,101)( 83,103)( 84,104)( 85,106)( 86,105)( 87,107)( 88,108)( 89,110)
( 90,109)( 91,111)( 92,112)( 93,114)( 94,113)( 95,115)( 96,116)( 97,118)
( 98,117)( 99,119)(100,120)(121,122)(125,126)(129,130)(133,134)(137,138)
(141,162)(142,161)(143,163)(144,164)(145,166)(146,165)(147,167)(148,168)
(149,170)(150,169)(151,171)(152,172)(153,174)(154,173)(155,175)(156,176)
(157,178)(158,177)(159,179)(160,180)(181,182)(185,186)(189,190)(193,194)
(197,198)(201,222)(202,221)(203,223)(204,224)(205,226)(206,225)(207,227)
(208,228)(209,230)(210,229)(211,231)(212,232)(213,234)(214,233)(215,235)
(216,236)(217,238)(218,237)(219,239)(220,240);
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3 >;

References : None.
to this polytope