Polytope of Type {4,45} with group SmallGroup(360,41)

Overview

Group: SmallGroup(360,41)
Rank: 3
Schläfli Type: {4,45}
Vertices, edges, …: 4, 90, 45
Order of s₀s₁s₂: 45
Order of s₀s₁s₂s₁: 4
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat

Quotients maximal quotients in bold

3-fold

{4,15}*120

5-fold

{4,9}*72

15-fold

{4,3}*24

Covers minimal covers in bold

2-fold

3-fold

{4,135}*1080

4-fold

5-fold

{4,225}*1800

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)( 15, 50)( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)(120,144);;
s2 := (  1,133)(  2,136)(  3,135)(  4,134)(  5,141)(  6,144)(  7,143)(  8,142)(  9,137)( 10,140)( 11,139)( 12,138)( 13,121)( 14,124)( 15,123)( 16,122)( 17,129)( 18,132)( 19,131)( 20,130)( 21,125)( 22,128)( 23,127)( 24,126)( 25,169)( 26,172)( 27,171)( 28,170)( 29,177)( 30,180)( 31,179)( 32,178)( 33,173)( 34,176)( 35,175)( 36,174)( 37,157)( 38,160)( 39,159)( 40,158)( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)( 57,149)( 58,152)( 59,151)( 60,150)( 61, 73)( 62, 76)( 63, 75)( 64, 74)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)( 98,100)(101,105)(102,108)(103,107)(104,106);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(180)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)(177,179)(178,180);
s1 := Sym(180)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)( 15, 50)( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)(120,144);
s2 := Sym(180)!(  1,133)(  2,136)(  3,135)(  4,134)(  5,141)(  6,144)(  7,143)(  8,142)(  9,137)( 10,140)( 11,139)( 12,138)( 13,121)( 14,124)( 15,123)( 16,122)( 17,129)( 18,132)( 19,131)( 20,130)( 21,125)( 22,128)( 23,127)( 24,126)( 25,169)( 26,172)( 27,171)( 28,170)( 29,177)( 30,180)( 31,179)( 32,178)( 33,173)( 34,176)( 35,175)( 36,174)( 37,157)( 38,160)( 39,159)( 40,158)( 41,165)( 42,168)( 43,167)( 44,166)( 45,161)( 46,164)( 47,163)( 48,162)( 49,145)( 50,148)( 51,147)( 52,146)( 53,153)( 54,156)( 55,155)( 56,154)( 57,149)( 58,152)( 59,151)( 60,150)( 61, 73)( 62, 76)( 63, 75)( 64, 74)( 65, 81)( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 85,109)( 86,112)( 87,111)( 88,110)( 89,117)( 90,120)( 91,119)( 92,118)( 93,113)( 94,116)( 95,115)( 96,114)( 98,100)(101,105)(102,108)(103,107)(104,106);
poly := sub<Sym(180)|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*s2*s1*s0*s1*s2*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References