Polytope of Type {10,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6,10}*1200
Also Known As : {{10,6|2},{6,10|2}}. if this polytope has another name.
Group : SmallGroup(1200,1006)
Rank : 4
Schlafli Type : {10,6,10}
Number of vertices, edges, etc : 10, 30, 30, 10
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,2,10}*400
   5-fold quotients : {2,6,10}*240, {10,6,2}*240
   6-fold quotients : {5,2,10}*200, {10,2,5}*200
   12-fold quotients : {5,2,5}*100
   15-fold quotients : {2,2,10}*80, {10,2,2}*80
   25-fold quotients : {2,6,2}*48
   30-fold quotients : {2,2,5}*40, {5,2,2}*40
   50-fold quotients : {2,3,2}*24
   75-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
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