Part of the Atlas of Small Regular Polytopes

Polytope of Type {42,3}

Atlas Canonical Name {42,3}*1764

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Overview

Group
SmallGroup(1764,146)
Rank
3
Schläfli Type
{42,3}
Vertices, edges, …
294, 441, 21
Order of s0s1s2
6
Order of s0s1s2s1
42
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

49-fold

147-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^6> of order 7

9 facets

42 vertex figures

P/N, where N=<(s0*s1)^7*s0*s2*(s1*s0)^2*s2*s1> of order 7

3 facets

42 vertex figures

P/N, where N=<(s0*s1)^2*s0*s2*s1*s0*s1*s2> of order 7

3 facets

42 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 43)(  9, 49)( 10, 48)( 11, 47)( 12, 46)( 13, 45)( 14, 44)( 15, 36)( 16, 42)( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)( 56,100)( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)( 64,134)( 65,140)( 66,139)( 67,138)( 68,137)( 69,136)( 70,135)( 71,127)( 72,133)( 73,132)( 74,131)( 75,130)( 76,129)( 77,128)( 78,120)( 79,126)( 80,125)( 81,124)( 82,123)( 83,122)( 84,121)( 85,113)( 86,119)( 87,118)( 88,117)( 89,116)( 90,115)( 91,114)( 92,106)( 93,112)( 94,111)( 95,110)( 96,109)( 97,108)( 98,107);;
s1 := (  1, 58)(  2, 51)(  3, 93)(  4, 86)(  5, 79)(  6, 72)(  7, 65)(  8, 57)(  9, 50)( 10, 92)( 11, 85)( 12, 78)( 13, 71)( 14, 64)( 15, 63)( 16, 56)( 17, 98)( 18, 91)( 19, 84)( 20, 77)( 21, 70)( 22, 62)( 23, 55)( 24, 97)( 25, 90)( 26, 83)( 27, 76)( 28, 69)( 29, 61)( 30, 54)( 31, 96)( 32, 89)( 33, 82)( 34, 75)( 35, 68)( 36, 60)( 37, 53)( 38, 95)( 39, 88)( 40, 81)( 41, 74)( 42, 67)( 43, 59)( 44, 52)( 45, 94)( 46, 87)( 47, 80)( 48, 73)( 49, 66)( 99,107)(101,142)(102,135)(103,128)(104,121)(105,114)(108,141)(109,134)(110,127)(111,120)(112,113)(115,147)(116,140)(117,133)(118,126)(122,146)(123,139)(124,132)(129,145)(130,138)(136,144);;
s2 := (  2, 36)(  3, 22)(  4,  8)(  5, 43)(  6, 29)(  7, 15)(  9, 39)( 10, 25)( 12, 46)( 13, 32)( 14, 18)( 16, 42)( 17, 28)( 19, 49)( 20, 35)( 23, 38)( 26, 45)( 27, 31)( 30, 41)( 33, 48)( 40, 44)( 50, 99)( 51,134)( 52,120)( 53,106)( 54,141)( 55,127)( 56,113)( 57,102)( 58,137)( 59,123)( 60,109)( 61,144)( 62,130)( 63,116)( 64,105)( 65,140)( 66,126)( 67,112)( 68,147)( 69,133)( 70,119)( 71,101)( 72,136)( 73,122)( 74,108)( 75,143)( 76,129)( 77,115)( 78,104)( 79,139)( 80,125)( 81,111)( 82,146)( 83,132)( 84,118)( 85,100)( 86,135)( 87,121)( 88,107)( 89,142)( 90,128)( 91,114)( 92,103)( 93,138)( 94,124)( 95,110)( 96,145)( 97,131)( 98,117);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(147)!(  2,  7)(  3,  6)(  4,  5)(  8, 43)(  9, 49)( 10, 48)( 11, 47)( 12, 46)( 13, 45)( 14, 44)( 15, 36)( 16, 42)( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 29)( 23, 35)( 24, 34)( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 50, 99)( 51,105)( 52,104)( 53,103)( 54,102)( 55,101)( 56,100)( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)( 64,134)( 65,140)( 66,139)( 67,138)( 68,137)( 69,136)( 70,135)( 71,127)( 72,133)( 73,132)( 74,131)( 75,130)( 76,129)( 77,128)( 78,120)( 79,126)( 80,125)( 81,124)( 82,123)( 83,122)( 84,121)( 85,113)( 86,119)( 87,118)( 88,117)( 89,116)( 90,115)( 91,114)( 92,106)( 93,112)( 94,111)( 95,110)( 96,109)( 97,108)( 98,107);
s1 := Sym(147)!(  1, 58)(  2, 51)(  3, 93)(  4, 86)(  5, 79)(  6, 72)(  7, 65)(  8, 57)(  9, 50)( 10, 92)( 11, 85)( 12, 78)( 13, 71)( 14, 64)( 15, 63)( 16, 56)( 17, 98)( 18, 91)( 19, 84)( 20, 77)( 21, 70)( 22, 62)( 23, 55)( 24, 97)( 25, 90)( 26, 83)( 27, 76)( 28, 69)( 29, 61)( 30, 54)( 31, 96)( 32, 89)( 33, 82)( 34, 75)( 35, 68)( 36, 60)( 37, 53)( 38, 95)( 39, 88)( 40, 81)( 41, 74)( 42, 67)( 43, 59)( 44, 52)( 45, 94)( 46, 87)( 47, 80)( 48, 73)( 49, 66)( 99,107)(101,142)(102,135)(103,128)(104,121)(105,114)(108,141)(109,134)(110,127)(111,120)(112,113)(115,147)(116,140)(117,133)(118,126)(122,146)(123,139)(124,132)(129,145)(130,138)(136,144);
s2 := Sym(147)!(  2, 36)(  3, 22)(  4,  8)(  5, 43)(  6, 29)(  7, 15)(  9, 39)( 10, 25)( 12, 46)( 13, 32)( 14, 18)( 16, 42)( 17, 28)( 19, 49)( 20, 35)( 23, 38)( 26, 45)( 27, 31)( 30, 41)( 33, 48)( 40, 44)( 50, 99)( 51,134)( 52,120)( 53,106)( 54,141)( 55,127)( 56,113)( 57,102)( 58,137)( 59,123)( 60,109)( 61,144)( 62,130)( 63,116)( 64,105)( 65,140)( 66,126)( 67,112)( 68,147)( 69,133)( 70,119)( 71,101)( 72,136)( 73,122)( 74,108)( 75,143)( 76,129)( 77,115)( 78,104)( 79,139)( 80,125)( 81,111)( 82,146)( 83,132)( 84,118)( 85,100)( 86,135)( 87,121)( 88,107)( 89,142)( 90,128)( 91,114)( 92,103)( 93,138)( 94,124)( 95,110)( 96,145)( 97,131)( 98,117);
poly := sub<Sym(147)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

Twisty Puzzle