Polytope of Type {3,4,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,10}*480
Also Known As : {{3,4},{4,10|2}}. if this polytope has another name.
Group : SmallGroup(480,1193)
Rank : 4
Schlafli Type : {3,4,10}
Number of vertices, edges, etc : 6, 12, 40, 10
Order of s0s1s2s3 : 30
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,4,10,2} of size 960
{3,4,10,4} of size 1920
Vertex Figure Of :
{2,3,4,10} of size 960
{3,3,4,10} of size 1920
{4,3,4,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
4-fold quotients : {3,2,10}*120
5-fold quotients : {3,4,2}*96
8-fold quotients : {3,2,5}*60
10-fold quotients : {3,4,2}*48
20-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,4,20}*960, {3,8,10}*960, {6,4,10}*960
3-fold covers : {9,4,10}*1440, {3,12,10}*1440, {3,4,30}*1440
4-fold covers : {3,8,20}*1920, {3,4,20}*1920, {3,8,10}*1920, {3,4,40}*1920, {12,4,10}*1920b, {6,4,20}*1920b, {6,4,10}*1920, {12,4,10}*1920c, {6,8,10}*1920a, {6,8,10}*1920b
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s2> of order 2.
10 facets:
10 of 2-fold non-regular quotient of {3,4}*48
4 vertex figures:
2 of {2,10}*40
2 of {4,10}*80
Permutation Representation (GAP) :
s0 := ( 2, 3)( 6, 7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)( 99,118)(100,120);;
s1 := ( 1, 21)( 2, 22)( 3, 24)( 4, 23)( 5, 25)( 6, 26)( 7, 28)( 8, 27)( 9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)( 67, 88)( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)(111,112)(115,116)(119,120);;
s2 := ( 1, 4)( 2, 3)( 5, 20)( 6, 19)( 7, 18)( 8, 17)( 9, 16)( 10, 15)( 11, 14)( 12, 13)( 21, 24)( 22, 23)( 25, 40)( 26, 39)( 27, 38)( 28, 37)( 29, 36)( 30, 35)( 31, 34)( 32, 33)( 41, 44)( 42, 43)( 45, 60)( 46, 59)( 47, 58)( 48, 57)( 49, 56)( 50, 55)( 51, 54)( 52, 53)( 61, 64)( 62, 63)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 92, 93)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113);;
s3 := ( 1, 65)( 2, 66)( 3, 67)( 4, 68)( 5, 61)( 6, 62)( 7, 63)( 8, 64)( 9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 73)( 14, 74)( 15, 75)( 16, 76)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 85)( 22, 86)( 23, 87)( 24, 88)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 97)( 30, 98)( 31, 99)( 32,100)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)( 40, 92)( 41,105)( 42,106)( 43,107)( 44,108)( 45,101)( 46,102)( 47,103)( 48,104)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)( 57,109)( 58,110)( 59,111)( 60,112);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
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(120)!( 2, 3)( 6, 7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)( 99,118)(100,120);
s1 := Sym(120)!( 1, 21)( 2, 22)( 3, 24)( 4, 23)( 5, 25)( 6, 26)( 7, 28)( 8, 27)( 9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)( 16, 35)( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)( 67, 88)( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)( 75, 96)( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)(111,112)(115,116)(119,120);
s2 := Sym(120)!( 1, 4)( 2, 3)( 5, 20)( 6, 19)( 7, 18)( 8, 17)( 9, 16)( 10, 15)( 11, 14)( 12, 13)( 21, 24)( 22, 23)( 25, 40)( 26, 39)( 27, 38)( 28, 37)( 29, 36)( 30, 35)( 31, 34)( 32, 33)( 41, 44)( 42, 43)( 45, 60)( 46, 59)( 47, 58)( 48, 57)( 49, 56)( 50, 55)( 51, 54)( 52, 53)( 61, 64)( 62, 63)( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 81, 84)( 82, 83)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 92, 93)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113);
s3 := Sym(120)!( 1, 65)( 2, 66)( 3, 67)( 4, 68)( 5, 61)( 6, 62)( 7, 63)( 8, 64)( 9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 73)( 14, 74)( 15, 75)( 16, 76)( 17, 69)( 18, 70)( 19, 71)( 20, 72)( 21, 85)( 22, 86)( 23, 87)( 24, 88)( 25, 81)( 26, 82)( 27, 83)( 28, 84)( 29, 97)( 30, 98)( 31, 99)( 32,100)( 33, 93)( 34, 94)( 35, 95)( 36, 96)( 37, 89)( 38, 90)( 39, 91)( 40, 92)( 41,105)( 42,106)( 43,107)( 44,108)( 45,101)( 46,102)( 47,103)( 48,104)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)( 57,109)( 58,110)( 59,111)( 60,112);
poly := sub<Sym(120)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, 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