Polytope of Type {6,10,10} with group SmallGroup(1200,1006)

Overview

Group: SmallGroup(1200,1006)
Rank: 4
Schläfli Type: {6,10,10}
Vertices, edges, …: 6, 30, 50, 10
Order of s₀s₁s₂s₃: 30
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: {{6,10|2},{10,10|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

3-fold

{2,10,10}*400a

5-fold

10-fold

15-fold

20-fold

{3,2,5}*60

25-fold

{6,2,2}*48

30-fold

50-fold

{3,2,2}*24

75-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := ( 26, 51)( 27, 52)( 28, 53)( 29, 54)( 30, 55)( 31, 56)( 32, 57)( 33, 58)( 34, 59)( 35, 60)( 36, 61)( 37, 62)( 38, 63)( 39, 64)( 40, 65)( 41, 66)( 42, 67)( 43, 68)( 44, 69)( 45, 70)( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150);;
s1 := (  1, 26)(  2, 27)(  3, 28)(  4, 29)(  5, 30)(  6, 46)(  7, 47)(  8, 48)(  9, 49)( 10, 50)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)( 16, 36)( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 76,101)( 77,102)( 78,103)( 79,104)( 80,105)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,116)( 87,117)( 88,118)( 89,119)( 90,120)( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)( 96,106)( 97,107)( 98,108)( 99,109)(100,110)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)(137,142)(138,143)(139,144)(140,145);;
s2 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)( 30, 32)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 42, 45)( 43, 44)( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)( 67, 70)( 68, 69)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)( 93, 94)(101,106)(102,110)(103,109)(104,108)(105,107)(111,121)(112,125)(113,124)(114,123)(115,122)(117,120)(118,119)(126,131)(127,135)(128,134)(129,133)(130,132)(136,146)(137,150)(138,149)(139,148)(140,147)(142,145)(143,144);;
s3 := (  1, 77)(  2, 76)(  3, 80)(  4, 79)(  5, 78)(  6, 82)(  7, 81)(  8, 85)(  9, 84)( 10, 83)( 11, 87)( 12, 86)( 13, 90)( 14, 89)( 15, 88)( 16, 92)( 17, 91)( 18, 95)( 19, 94)( 20, 93)( 21, 97)( 22, 96)( 23,100)( 24, 99)( 25, 98)( 26,102)( 27,101)( 28,105)( 29,104)( 30,103)( 31,107)( 32,106)( 33,110)( 34,109)( 35,108)( 36,112)( 37,111)( 38,115)( 39,114)( 40,113)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46,122)( 47,121)( 48,125)( 49,124)( 50,123)( 51,127)( 52,126)( 53,130)( 54,129)( 55,128)( 56,132)( 57,131)( 58,135)( 59,134)( 60,133)( 61,137)( 62,136)( 63,140)( 64,139)( 65,138)( 66,142)( 67,141)( 68,145)( 69,144)( 70,143)( 71,147)( 72,146)( 73,150)( 74,149)( 75,148);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(150)!( 26, 51)( 27, 52)( 28, 53)( 29, 54)( 30, 55)( 31, 56)( 32, 57)( 33, 58)( 34, 59)( 35, 60)( 36, 61)( 37, 62)( 38, 63)( 39, 64)( 40, 65)( 41, 66)( 42, 67)( 43, 68)( 44, 69)( 45, 70)( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150);
s1 := Sym(150)!(  1, 26)(  2, 27)(  3, 28)(  4, 29)(  5, 30)(  6, 46)(  7, 47)(  8, 48)(  9, 49)( 10, 50)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)( 16, 36)( 17, 37)( 18, 38)( 19, 39)( 20, 40)( 21, 31)( 22, 32)( 23, 33)( 24, 34)( 25, 35)( 56, 71)( 57, 72)( 58, 73)( 59, 74)( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 76,101)( 77,102)( 78,103)( 79,104)( 80,105)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,116)( 87,117)( 88,118)( 89,119)( 90,120)( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)( 96,106)( 97,107)( 98,108)( 99,109)(100,110)(131,146)(132,147)(133,148)(134,149)(135,150)(136,141)(137,142)(138,143)(139,144)(140,145);
s2 := Sym(150)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 17, 20)( 18, 19)( 26, 31)( 27, 35)( 28, 34)( 29, 33)( 30, 32)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)( 42, 45)( 43, 44)( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)( 67, 70)( 68, 69)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)( 86, 96)( 87,100)( 88, 99)( 89, 98)( 90, 97)( 92, 95)( 93, 94)(101,106)(102,110)(103,109)(104,108)(105,107)(111,121)(112,125)(113,124)(114,123)(115,122)(117,120)(118,119)(126,131)(127,135)(128,134)(129,133)(130,132)(136,146)(137,150)(138,149)(139,148)(140,147)(142,145)(143,144);
s3 := Sym(150)!(  1, 77)(  2, 76)(  3, 80)(  4, 79)(  5, 78)(  6, 82)(  7, 81)(  8, 85)(  9, 84)( 10, 83)( 11, 87)( 12, 86)( 13, 90)( 14, 89)( 15, 88)( 16, 92)( 17, 91)( 18, 95)( 19, 94)( 20, 93)( 21, 97)( 22, 96)( 23,100)( 24, 99)( 25, 98)( 26,102)( 27,101)( 28,105)( 29,104)( 30,103)( 31,107)( 32,106)( 33,110)( 34,109)( 35,108)( 36,112)( 37,111)( 38,115)( 39,114)( 40,113)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46,122)( 47,121)( 48,125)( 49,124)( 50,123)( 51,127)( 52,126)( 53,130)( 54,129)( 55,128)( 56,132)( 57,131)( 58,135)( 59,134)( 60,133)( 61,137)( 62,136)( 63,140)( 64,139)( 65,138)( 66,142)( 67,141)( 68,145)( 69,144)( 70,143)( 71,147)( 72,146)( 73,150)( 74,149)( 75,148);
poly := sub<Sym(150)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References