Polytope of Type {2,40,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,40,6}*1920b
if this polytope has a name.
Group : SmallGroup(1920,240174)
Rank : 4
Schlafli Type : {2,40,6}
Number of vertices, edges, etc : 2, 80, 240, 12
Order of s0s1s2s3 : 30
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 : {2,20,6}*960c
4-fold quotients : {2,20,6}*480b
5-fold quotients : {2,8,6}*384c
8-fold quotients : {2,10,6}*240
10-fold quotients : {2,4,6}*192
20-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
24-fold quotients : {2,10,2}*80
40-fold quotients : {2,4,3}*48, {2,2,6}*48
48-fold quotients : {2,5,2}*40
80-fold quotients : {2,2,3}*24
120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,127)( 4,128)( 5,130)( 6,129)( 7,124)( 8,123)( 9,125)( 10,126)
( 11,159)( 12,160)( 13,162)( 14,161)( 15,156)( 16,155)( 17,157)( 18,158)
( 19,151)( 20,152)( 21,154)( 22,153)( 23,148)( 24,147)( 25,149)( 26,150)
( 27,143)( 28,144)( 29,146)( 30,145)( 31,140)( 32,139)( 33,141)( 34,142)
( 35,135)( 36,136)( 37,138)( 38,137)( 39,132)( 40,131)( 41,133)( 42,134)
( 43,167)( 44,168)( 45,170)( 46,169)( 47,164)( 48,163)( 49,165)( 50,166)
( 51,199)( 52,200)( 53,202)( 54,201)( 55,196)( 56,195)( 57,197)( 58,198)
( 59,191)( 60,192)( 61,194)( 62,193)( 63,188)( 64,187)( 65,189)( 66,190)
( 67,183)( 68,184)( 69,186)( 70,185)( 71,180)( 72,179)( 73,181)( 74,182)
( 75,175)( 76,176)( 77,178)( 78,177)( 79,172)( 80,171)( 81,173)( 82,174)
( 83,207)( 84,208)( 85,210)( 86,209)( 87,204)( 88,203)( 89,205)( 90,206)
( 91,239)( 92,240)( 93,242)( 94,241)( 95,236)( 96,235)( 97,237)( 98,238)
( 99,231)(100,232)(101,234)(102,233)(103,228)(104,227)(105,229)(106,230)
(107,223)(108,224)(109,226)(110,225)(111,220)(112,219)(113,221)(114,222)
(115,215)(116,216)(117,218)(118,217)(119,212)(120,211)(121,213)(122,214);;
s2 := ( 3, 11)( 4, 12)( 5, 16)( 6, 15)( 7, 14)( 8, 13)( 9, 18)( 10, 17)
( 19, 35)( 20, 36)( 21, 40)( 22, 39)( 23, 38)( 24, 37)( 25, 42)( 26, 41)
( 29, 32)( 30, 31)( 33, 34)( 43, 91)( 44, 92)( 45, 96)( 46, 95)( 47, 94)
( 48, 93)( 49, 98)( 50, 97)( 51, 83)( 52, 84)( 53, 88)( 54, 87)( 55, 86)
( 56, 85)( 57, 90)( 58, 89)( 59,115)( 60,116)( 61,120)( 62,119)( 63,118)
( 64,117)( 65,122)( 66,121)( 67,107)( 68,108)( 69,112)( 70,111)( 71,110)
( 72,109)( 73,114)( 74,113)( 75, 99)( 76,100)( 77,104)( 78,103)( 79,102)
( 80,101)( 81,106)( 82,105)(123,131)(124,132)(125,136)(126,135)(127,134)
(128,133)(129,138)(130,137)(139,155)(140,156)(141,160)(142,159)(143,158)
(144,157)(145,162)(146,161)(149,152)(150,151)(153,154)(163,211)(164,212)
(165,216)(166,215)(167,214)(168,213)(169,218)(170,217)(171,203)(172,204)
(173,208)(174,207)(175,206)(176,205)(177,210)(178,209)(179,235)(180,236)
(181,240)(182,239)(183,238)(184,237)(185,242)(186,241)(187,227)(188,228)
(189,232)(190,231)(191,230)(192,229)(193,234)(194,233)(195,219)(196,220)
(197,224)(198,223)(199,222)(200,221)(201,226)(202,225);;
s3 := ( 3, 83)( 4, 84)( 5, 89)( 6, 90)( 7, 88)( 8, 87)( 9, 85)( 10, 86)
( 11, 91)( 12, 92)( 13, 97)( 14, 98)( 15, 96)( 16, 95)( 17, 93)( 18, 94)
( 19, 99)( 20,100)( 21,105)( 22,106)( 23,104)( 24,103)( 25,101)( 26,102)
( 27,107)( 28,108)( 29,113)( 30,114)( 31,112)( 32,111)( 33,109)( 34,110)
( 35,115)( 36,116)( 37,121)( 38,122)( 39,120)( 40,119)( 41,117)( 42,118)
( 45, 49)( 46, 50)( 47, 48)( 53, 57)( 54, 58)( 55, 56)( 61, 65)( 62, 66)
( 63, 64)( 69, 73)( 70, 74)( 71, 72)( 77, 81)( 78, 82)( 79, 80)(123,204)
(124,203)(125,210)(126,209)(127,207)(128,208)(129,206)(130,205)(131,212)
(132,211)(133,218)(134,217)(135,215)(136,216)(137,214)(138,213)(139,220)
(140,219)(141,226)(142,225)(143,223)(144,224)(145,222)(146,221)(147,228)
(148,227)(149,234)(150,233)(151,231)(152,232)(153,230)(154,229)(155,236)
(156,235)(157,242)(158,241)(159,239)(160,240)(161,238)(162,237)(163,164)
(165,170)(166,169)(171,172)(173,178)(174,177)(179,180)(181,186)(182,185)
(187,188)(189,194)(190,193)(195,196)(197,202)(198,201);;
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,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s2*s3*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(242)!(1,2);
s1 := Sym(242)!( 3,127)( 4,128)( 5,130)( 6,129)( 7,124)( 8,123)( 9,125)
( 10,126)( 11,159)( 12,160)( 13,162)( 14,161)( 15,156)( 16,155)( 17,157)
( 18,158)( 19,151)( 20,152)( 21,154)( 22,153)( 23,148)( 24,147)( 25,149)
( 26,150)( 27,143)( 28,144)( 29,146)( 30,145)( 31,140)( 32,139)( 33,141)
( 34,142)( 35,135)( 36,136)( 37,138)( 38,137)( 39,132)( 40,131)( 41,133)
( 42,134)( 43,167)( 44,168)( 45,170)( 46,169)( 47,164)( 48,163)( 49,165)
( 50,166)( 51,199)( 52,200)( 53,202)( 54,201)( 55,196)( 56,195)( 57,197)
( 58,198)( 59,191)( 60,192)( 61,194)( 62,193)( 63,188)( 64,187)( 65,189)
( 66,190)( 67,183)( 68,184)( 69,186)( 70,185)( 71,180)( 72,179)( 73,181)
( 74,182)( 75,175)( 76,176)( 77,178)( 78,177)( 79,172)( 80,171)( 81,173)
( 82,174)( 83,207)( 84,208)( 85,210)( 86,209)( 87,204)( 88,203)( 89,205)
( 90,206)( 91,239)( 92,240)( 93,242)( 94,241)( 95,236)( 96,235)( 97,237)
( 98,238)( 99,231)(100,232)(101,234)(102,233)(103,228)(104,227)(105,229)
(106,230)(107,223)(108,224)(109,226)(110,225)(111,220)(112,219)(113,221)
(114,222)(115,215)(116,216)(117,218)(118,217)(119,212)(120,211)(121,213)
(122,214);
s2 := Sym(242)!( 3, 11)( 4, 12)( 5, 16)( 6, 15)( 7, 14)( 8, 13)( 9, 18)
( 10, 17)( 19, 35)( 20, 36)( 21, 40)( 22, 39)( 23, 38)( 24, 37)( 25, 42)
( 26, 41)( 29, 32)( 30, 31)( 33, 34)( 43, 91)( 44, 92)( 45, 96)( 46, 95)
( 47, 94)( 48, 93)( 49, 98)( 50, 97)( 51, 83)( 52, 84)( 53, 88)( 54, 87)
( 55, 86)( 56, 85)( 57, 90)( 58, 89)( 59,115)( 60,116)( 61,120)( 62,119)
( 63,118)( 64,117)( 65,122)( 66,121)( 67,107)( 68,108)( 69,112)( 70,111)
( 71,110)( 72,109)( 73,114)( 74,113)( 75, 99)( 76,100)( 77,104)( 78,103)
( 79,102)( 80,101)( 81,106)( 82,105)(123,131)(124,132)(125,136)(126,135)
(127,134)(128,133)(129,138)(130,137)(139,155)(140,156)(141,160)(142,159)
(143,158)(144,157)(145,162)(146,161)(149,152)(150,151)(153,154)(163,211)
(164,212)(165,216)(166,215)(167,214)(168,213)(169,218)(170,217)(171,203)
(172,204)(173,208)(174,207)(175,206)(176,205)(177,210)(178,209)(179,235)
(180,236)(181,240)(182,239)(183,238)(184,237)(185,242)(186,241)(187,227)
(188,228)(189,232)(190,231)(191,230)(192,229)(193,234)(194,233)(195,219)
(196,220)(197,224)(198,223)(199,222)(200,221)(201,226)(202,225);
s3 := Sym(242)!( 3, 83)( 4, 84)( 5, 89)( 6, 90)( 7, 88)( 8, 87)( 9, 85)
( 10, 86)( 11, 91)( 12, 92)( 13, 97)( 14, 98)( 15, 96)( 16, 95)( 17, 93)
( 18, 94)( 19, 99)( 20,100)( 21,105)( 22,106)( 23,104)( 24,103)( 25,101)
( 26,102)( 27,107)( 28,108)( 29,113)( 30,114)( 31,112)( 32,111)( 33,109)
( 34,110)( 35,115)( 36,116)( 37,121)( 38,122)( 39,120)( 40,119)( 41,117)
( 42,118)( 45, 49)( 46, 50)( 47, 48)( 53, 57)( 54, 58)( 55, 56)( 61, 65)
( 62, 66)( 63, 64)( 69, 73)( 70, 74)( 71, 72)( 77, 81)( 78, 82)( 79, 80)
(123,204)(124,203)(125,210)(126,209)(127,207)(128,208)(129,206)(130,205)
(131,212)(132,211)(133,218)(134,217)(135,215)(136,216)(137,214)(138,213)
(139,220)(140,219)(141,226)(142,225)(143,223)(144,224)(145,222)(146,221)
(147,228)(148,227)(149,234)(150,233)(151,231)(152,232)(153,230)(154,229)
(155,236)(156,235)(157,242)(158,241)(159,239)(160,240)(161,238)(162,237)
(163,164)(165,170)(166,169)(171,172)(173,178)(174,177)(179,180)(181,186)
(182,185)(187,188)(189,194)(190,193)(195,196)(197,202)(198,201);
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,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s2*s3*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1 >;
to this polytope