Polytope of Type {36,8} with group SmallGroup(576,335)

Overview

Group: SmallGroup(576,335)
Rank: 3
Schläfli Type: {36,8}
Vertices, edges, …: 36, 144, 8
Order of s₀s₁s₂: 72
Order of s₀s₁s₂s₁: 2
Also known as: {36,8|2}. if this polytope has another name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat

Quotients maximal quotients in bold

2-fold

3-fold

{12,8}*192a

4-fold

6-fold

8-fold

{18,2}*72

9-fold

{4,8}*64a

12-fold

16-fold

{9,2}*36

18-fold

24-fold

{6,2}*24

36-fold

48-fold

{3,2}*12

72-fold

{2,2}*8

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References