Polytope of Type {78,4} with group SmallGroup(1872,1023)

Overview

Group: SmallGroup(1872,1023)
Rank: 3
Schläfli Type: {78,4}
Vertices, edges, …: 234, 468, 12
Order of s₀s₁s₂: 52
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

9-fold

{26,4}*208

13-fold

{6,4}*144

18-fold

{26,2}*104

26-fold

{6,4}*72

36-fold

{13,2}*52

117-fold

{2,4}*16

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)^8*s0*s2*(s1*s0)^3*s2*s1*s0*s1> of order 2

7 facets

5 of {78}*156
2 of {39}*78

117 vertex figures

117 of {4}*8

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

6 facets

6 of {78}*156

117 vertex figures

117 of {4}*8

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

8 facets

2 of {78}*156
6 of {26}*52

78 vertex figures

78 of {4}*8

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

4 facets

4 of {78}*156

78 vertex figures

78 of {4}*8

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, (s0*s1)^10*s0*s2*(s1*s0)^2*s1*s2> of order 6

5 facets

3 of {26}*52
2 of {39}*78

39 vertex figures

39 of {4}*8

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.