Part of the Atlas of Small Regular Polytopes

Polytope of Type {10,4}

Atlas Canonical Name {10,4}*1280c

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Overview

Group
SmallGroup(1280,1116459)
Rank
3
Schläfli Type
{10,4}
Vertices, edges, …
160, 320, 64
Order of s0s1s2
10
Order of s0s1s2s1
8
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Self-Petrie

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

32-fold

64-fold

160-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*s2*s1)^5> of order 2

32 facets

80 vertex figures

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

32 facets

80 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^3*s2*s1*s2> of order 2

32 facets

80 vertex figures

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

32 facets

88 vertex figures

P/N, where N=<(s0*s1)^5> of order 2

48 facets

80 vertex figures

P/N, where N=<(s1*s0)^2*s1*(s2*s1*s0)^2*s1, (s0*s1)^2*(s2*s1*s0)^2*s1*s2*s1> of order 4

16 facets

44 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2> of order 4

16 facets

44 vertex figures

P/N, where N=<(s1*s0)^3*s2*s1*s0*s1*s2> of order 4

16 facets

44 vertex figures

P/N, where N=<(s1*s0)^2*s1*(s2*s1*s0)^2*s1, s0*s1*s0*s2*(s1*s0)^3*s2*s1*s2> of order 4

16 facets

44 vertex figures

Representations

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

References

None.

to this polytope.

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