Polytope of Type {52,4} with group SmallGroup(1872,1023)

Overview

Group: SmallGroup(1872,1023)
Rank: 3
Schläfli Type: {52,4}
Vertices, edges, …: 234, 468, 18
Order of s₀s₁s₂: 78
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

13-fold

{4,4}*144

18-fold

{26,2}*104

26-fold

{4,4}*72

36-fold

{13,2}*52

234-fold

{2,2}*8

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)^9*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 2

9 facets

9 of {52}*104

117 vertex figures

117 of {4}*8

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

9 facets

9 of {52}*104

117 vertex figures

117 of {4}*8

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

10 facets

2 of {26}*52
8 of {52}*104

130 vertex figures

104 of {4}*8
26 of {2}*4

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

6 facets

6 of {52}*104

78 vertex figures

78 of {4}*8

P/N, where N=<(s2*s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1*s2, (s0*s1)^9*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 4

5 facets

1 of {26}*52
4 of {52}*104

65 vertex figures

52 of {4}*8
13 of {2}*4

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 6

4 facets

2 of {26}*52
2 of {52}*104

52 vertex figures

26 of {4}*8
26 of {2}*4

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.