Polytope of Type {6,6} with group SmallGroup(576,8328)

Overview

Group: SmallGroup(576,8328)
Rank: 3
Schläfli Type: {6,6}
Vertices, edges, …: 48, 144, 48
Order of s₀s₁s₂: 24
Order of s₀s₁s₂s₁: 12
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{6,6}*288a

3-fold

{6,6}*192b

4-fold

{6,3}*144

6-fold

{6,6}*96

8-fold

{6,6}*72b

12-fold

16-fold

{6,3}*36

24-fold

48-fold

{2,3}*12

72-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

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

24 facets

24 of {6}*12

36 vertex figures

12 of {6}*12
24 of {3}*6

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

28 facets

8 of {3}*6
20 of {6}*12

24 vertex figures

24 of {6}*12

P/N, where N=<s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 3

16 facets

16 of {6}*12

20 vertex figures

14 of {6}*12
6 of {2}*4

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

24 facets

12 of {2}*4
12 of {6}*12

16 vertex figures

16 of {6}*12

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

12 facets

6 of {2}*4
6 of {6}*12

12 vertex figures

4 of {6}*12
8 of {3}*6

P/N, where N=<(s1*s2)^3, (s1*s2*s1*s0)^2*s1*s2> of order 6

8 facets

8 of {6}*12

14 vertex figures

8 of {3}*6
3 of {6}*12
3 of {2}*4

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.