Polytope of Type {2,30,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,30,6}*1920
if this polytope has a name.
Group : SmallGroup(1920,240174)
Rank : 4
Schlafli Type : {2,30,6}
Number of vertices, edges, etc : 2, 80, 240, 16
Order of s0s1s2s3 : 40
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,6,6}*384b
10-fold quotients : {2,6,6}*192
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
120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,123)( 4,124)( 5,128)( 6,127)( 7,126)( 8,125)( 9,130)( 10,129)
( 11,155)( 12,156)( 13,160)( 14,159)( 15,158)( 16,157)( 17,162)( 18,161)
( 19,147)( 20,148)( 21,152)( 22,151)( 23,150)( 24,149)( 25,154)( 26,153)
( 27,139)( 28,140)( 29,144)( 30,143)( 31,142)( 32,141)( 33,146)( 34,145)
( 35,131)( 36,132)( 37,136)( 38,135)( 39,134)( 40,133)( 41,138)( 42,137)
( 43,203)( 44,204)( 45,208)( 46,207)( 47,206)( 48,205)( 49,210)( 50,209)
( 51,235)( 52,236)( 53,240)( 54,239)( 55,238)( 56,237)( 57,242)( 58,241)
( 59,227)( 60,228)( 61,232)( 62,231)( 63,230)( 64,229)( 65,234)( 66,233)
( 67,219)( 68,220)( 69,224)( 70,223)( 71,222)( 72,221)( 73,226)( 74,225)
( 75,211)( 76,212)( 77,216)( 78,215)( 79,214)( 80,213)( 81,218)( 82,217)
( 83,163)( 84,164)( 85,168)( 86,167)( 87,166)( 88,165)( 89,170)( 90,169)
( 91,195)( 92,196)( 93,200)( 94,199)( 95,198)( 96,197)( 97,202)( 98,201)
( 99,187)(100,188)(101,192)(102,191)(103,190)(104,189)(105,194)(106,193)
(107,179)(108,180)(109,184)(110,183)(111,182)(112,181)(113,186)(114,185)
(115,171)(116,172)(117,176)(118,175)(119,174)(120,173)(121,178)(122,177);;
s2 := ( 3, 51)( 4, 52)( 5, 54)( 6, 53)( 7, 57)( 8, 58)( 9, 55)( 10, 56)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 49)( 16, 50)( 17, 47)( 18, 48)
( 19, 75)( 20, 76)( 21, 78)( 22, 77)( 23, 81)( 24, 82)( 25, 79)( 26, 80)
( 27, 67)( 28, 68)( 29, 70)( 30, 69)( 31, 73)( 32, 74)( 33, 71)( 34, 72)
( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 39, 65)( 40, 66)( 41, 63)( 42, 64)
( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 97)( 88, 98)( 89, 95)( 90, 96)
( 99,115)(100,116)(101,118)(102,117)(103,121)(104,122)(105,119)(106,120)
(109,110)(111,113)(112,114)(123,171)(124,172)(125,174)(126,173)(127,177)
(128,178)(129,175)(130,176)(131,163)(132,164)(133,166)(134,165)(135,169)
(136,170)(137,167)(138,168)(139,195)(140,196)(141,198)(142,197)(143,201)
(144,202)(145,199)(146,200)(147,187)(148,188)(149,190)(150,189)(151,193)
(152,194)(153,191)(154,192)(155,179)(156,180)(157,182)(158,181)(159,185)
(160,186)(161,183)(162,184)(203,211)(204,212)(205,214)(206,213)(207,217)
(208,218)(209,215)(210,216)(219,235)(220,236)(221,238)(222,237)(223,241)
(224,242)(225,239)(226,240)(229,230)(231,233)(232,234);;
s3 := ( 3, 9)( 4, 10)( 7, 8)( 11, 17)( 12, 18)( 15, 16)( 19, 25)( 20, 26)
( 23, 24)( 27, 33)( 28, 34)( 31, 32)( 35, 41)( 36, 42)( 39, 40)( 43, 89)
( 44, 90)( 45, 85)( 46, 86)( 47, 88)( 48, 87)( 49, 83)( 50, 84)( 51, 97)
( 52, 98)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 91)( 58, 92)( 59,105)
( 60,106)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 99)( 66,100)( 67,113)
( 68,114)( 69,109)( 70,110)( 71,112)( 72,111)( 73,107)( 74,108)( 75,121)
( 76,122)( 77,117)( 78,118)( 79,120)( 80,119)( 81,115)( 82,116)(123,130)
(124,129)(125,126)(131,138)(132,137)(133,134)(139,146)(140,145)(141,142)
(147,154)(148,153)(149,150)(155,162)(156,161)(157,158)(163,210)(164,209)
(165,206)(166,205)(167,207)(168,208)(169,204)(170,203)(171,218)(172,217)
(173,214)(174,213)(175,215)(176,216)(177,212)(178,211)(179,226)(180,225)
(181,222)(182,221)(183,223)(184,224)(185,220)(186,219)(187,234)(188,233)
(189,230)(190,229)(191,231)(192,232)(193,228)(194,227)(195,242)(196,241)
(197,238)(198,237)(199,239)(200,240)(201,236)(202,235);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(242)!(1,2);
s1 := Sym(242)!( 3,123)( 4,124)( 5,128)( 6,127)( 7,126)( 8,125)( 9,130)
( 10,129)( 11,155)( 12,156)( 13,160)( 14,159)( 15,158)( 16,157)( 17,162)
( 18,161)( 19,147)( 20,148)( 21,152)( 22,151)( 23,150)( 24,149)( 25,154)
( 26,153)( 27,139)( 28,140)( 29,144)( 30,143)( 31,142)( 32,141)( 33,146)
( 34,145)( 35,131)( 36,132)( 37,136)( 38,135)( 39,134)( 40,133)( 41,138)
( 42,137)( 43,203)( 44,204)( 45,208)( 46,207)( 47,206)( 48,205)( 49,210)
( 50,209)( 51,235)( 52,236)( 53,240)( 54,239)( 55,238)( 56,237)( 57,242)
( 58,241)( 59,227)( 60,228)( 61,232)( 62,231)( 63,230)( 64,229)( 65,234)
( 66,233)( 67,219)( 68,220)( 69,224)( 70,223)( 71,222)( 72,221)( 73,226)
( 74,225)( 75,211)( 76,212)( 77,216)( 78,215)( 79,214)( 80,213)( 81,218)
( 82,217)( 83,163)( 84,164)( 85,168)( 86,167)( 87,166)( 88,165)( 89,170)
( 90,169)( 91,195)( 92,196)( 93,200)( 94,199)( 95,198)( 96,197)( 97,202)
( 98,201)( 99,187)(100,188)(101,192)(102,191)(103,190)(104,189)(105,194)
(106,193)(107,179)(108,180)(109,184)(110,183)(111,182)(112,181)(113,186)
(114,185)(115,171)(116,172)(117,176)(118,175)(119,174)(120,173)(121,178)
(122,177);
s2 := Sym(242)!( 3, 51)( 4, 52)( 5, 54)( 6, 53)( 7, 57)( 8, 58)( 9, 55)
( 10, 56)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 49)( 16, 50)( 17, 47)
( 18, 48)( 19, 75)( 20, 76)( 21, 78)( 22, 77)( 23, 81)( 24, 82)( 25, 79)
( 26, 80)( 27, 67)( 28, 68)( 29, 70)( 30, 69)( 31, 73)( 32, 74)( 33, 71)
( 34, 72)( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 39, 65)( 40, 66)( 41, 63)
( 42, 64)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 97)( 88, 98)( 89, 95)
( 90, 96)( 99,115)(100,116)(101,118)(102,117)(103,121)(104,122)(105,119)
(106,120)(109,110)(111,113)(112,114)(123,171)(124,172)(125,174)(126,173)
(127,177)(128,178)(129,175)(130,176)(131,163)(132,164)(133,166)(134,165)
(135,169)(136,170)(137,167)(138,168)(139,195)(140,196)(141,198)(142,197)
(143,201)(144,202)(145,199)(146,200)(147,187)(148,188)(149,190)(150,189)
(151,193)(152,194)(153,191)(154,192)(155,179)(156,180)(157,182)(158,181)
(159,185)(160,186)(161,183)(162,184)(203,211)(204,212)(205,214)(206,213)
(207,217)(208,218)(209,215)(210,216)(219,235)(220,236)(221,238)(222,237)
(223,241)(224,242)(225,239)(226,240)(229,230)(231,233)(232,234);
s3 := Sym(242)!( 3, 9)( 4, 10)( 7, 8)( 11, 17)( 12, 18)( 15, 16)( 19, 25)
( 20, 26)( 23, 24)( 27, 33)( 28, 34)( 31, 32)( 35, 41)( 36, 42)( 39, 40)
( 43, 89)( 44, 90)( 45, 85)( 46, 86)( 47, 88)( 48, 87)( 49, 83)( 50, 84)
( 51, 97)( 52, 98)( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 91)( 58, 92)
( 59,105)( 60,106)( 61,101)( 62,102)( 63,104)( 64,103)( 65, 99)( 66,100)
( 67,113)( 68,114)( 69,109)( 70,110)( 71,112)( 72,111)( 73,107)( 74,108)
( 75,121)( 76,122)( 77,117)( 78,118)( 79,120)( 80,119)( 81,115)( 82,116)
(123,130)(124,129)(125,126)(131,138)(132,137)(133,134)(139,146)(140,145)
(141,142)(147,154)(148,153)(149,150)(155,162)(156,161)(157,158)(163,210)
(164,209)(165,206)(166,205)(167,207)(168,208)(169,204)(170,203)(171,218)
(172,217)(173,214)(174,213)(175,215)(176,216)(177,212)(178,211)(179,226)
(180,225)(181,222)(182,221)(183,223)(184,224)(185,220)(186,219)(187,234)
(188,233)(189,230)(190,229)(191,231)(192,232)(193,228)(194,227)(195,242)
(196,241)(197,238)(198,237)(199,239)(200,240)(201,236)(202,235);
poly := sub<Sym(242)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2,
s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2 >;
to this polytope