Polytope of Type {6,10}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10}*600b
if this polytope has a name.
Group : SmallGroup(600,154)
Rank : 3
Schlafli Type : {6,10}
Number of vertices, edges, etc : 30, 150, 50
Order of s0s1s2 : 6
Order of s0s1s2s1 : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,10,2} of size 1200
Vertex Figure Of :
   {2,6,10} of size 1200
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,10}*300
   25-fold quotients : {6,2}*24
   50-fold quotients : {3,2}*12
   75-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,10}*1200a, {6,20}*1200b
   3-fold covers : {18,10}*1800b, {6,30}*1800a, {6,30}*1800d
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 5.
      10 facets:
         10 of {6}*12
      6 vertex figures:
         6 of {10}*20
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 5.
      10 facets:
         10 of {6}*12
      14 vertex figures:
         4 of {10}*20
         10 of {2}*4

Permutation Representation (GAP) : 
s0 := (  2,  7)(  3, 13)(  4, 19)(  5, 25)(  6, 21)(  9, 14)( 10, 20)( 11, 16)( 12, 22)( 18, 23)( 26, 51)( 27, 57)( 28, 63)( 29, 69)( 30, 75)( 31, 71)( 32, 52)( 33, 58)( 34, 64)( 35, 70)( 36, 66)( 37, 72)( 38, 53)( 39, 59)( 40, 65)( 41, 61)( 42, 67)( 43, 73)( 44, 54)( 45, 60)( 46, 56)( 47, 62)( 48, 68)( 49, 74)( 50, 55)( 77, 82)( 78, 88)( 79, 94)( 80,100)( 81, 96)( 84, 89)( 85, 95)( 86, 91)( 87, 97)( 93, 98)(101,126)(102,132)(103,138)(104,144)(105,150)(106,146)(107,127)(108,133)(109,139)(110,145)(111,141)(112,147)(113,128)(114,134)(115,140)(116,136)(117,142)(118,148)(119,129)(120,135)(121,131)(122,137)(123,143)(124,149)(125,130);;
s1 := (  1,101)(  2,124)(  3,117)(  4,115)(  5,108)(  6,114)(  7,107)(  8,105)(  9,123)( 10,116)( 11,122)( 12,120)( 13,113)( 14,106)( 15,104)( 16,110)( 17,103)( 18,121)( 19,119)( 20,112)( 21,118)( 22,111)( 23,109)( 24,102)( 25,125)( 26, 76)( 27, 99)( 28, 92)( 29, 90)( 30, 83)( 31, 89)( 32, 82)( 33, 80)( 34, 98)( 35, 91)( 36, 97)( 37, 95)( 38, 88)( 39, 81)( 40, 79)( 41, 85)( 42, 78)( 43, 96)( 44, 94)( 45, 87)( 46, 93)( 47, 86)( 48, 84)( 49, 77)( 50,100)( 51,126)( 52,149)( 53,142)( 54,140)( 55,133)( 56,139)( 57,132)( 58,130)( 59,148)( 60,141)( 61,147)( 62,145)( 63,138)( 64,131)( 65,129)( 66,135)( 67,128)( 68,146)( 69,144)( 70,137)( 71,143)( 72,136)( 73,134)( 74,127)( 75,150);;
s2 := (  1,  8)(  2,  7)(  3,  6)(  4, 10)(  5,  9)( 11, 23)( 12, 22)( 13, 21)( 14, 25)( 15, 24)( 16, 18)( 19, 20)( 26, 33)( 27, 32)( 28, 31)( 29, 35)( 30, 34)( 36, 48)( 37, 47)( 38, 46)( 39, 50)( 40, 49)( 41, 43)( 44, 45)( 51, 58)( 52, 57)( 53, 56)( 54, 60)( 55, 59)( 61, 73)( 62, 72)( 63, 71)( 64, 75)( 65, 74)( 66, 68)( 69, 70)( 76, 83)( 77, 82)( 78, 81)( 79, 85)( 80, 84)( 86, 98)( 87, 97)( 88, 96)( 89,100)( 90, 99)( 91, 93)( 94, 95)(101,108)(102,107)(103,106)(104,110)(105,109)(111,123)(112,122)(113,121)(114,125)(115,124)(116,118)(119,120)(126,133)(127,132)(128,131)(129,135)(130,134)(136,148)(137,147)(138,146)(139,150)(140,149)(141,143)(144,145);;
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, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(150)!(  2,  7)(  3, 13)(  4, 19)(  5, 25)(  6, 21)(  9, 14)( 10, 20)( 11, 16)( 12, 22)( 18, 23)( 26, 51)( 27, 57)( 28, 63)( 29, 69)( 30, 75)( 31, 71)( 32, 52)( 33, 58)( 34, 64)( 35, 70)( 36, 66)( 37, 72)( 38, 53)( 39, 59)( 40, 65)( 41, 61)( 42, 67)( 43, 73)( 44, 54)( 45, 60)( 46, 56)( 47, 62)( 48, 68)( 49, 74)( 50, 55)( 77, 82)( 78, 88)( 79, 94)( 80,100)( 81, 96)( 84, 89)( 85, 95)( 86, 91)( 87, 97)( 93, 98)(101,126)(102,132)(103,138)(104,144)(105,150)(106,146)(107,127)(108,133)(109,139)(110,145)(111,141)(112,147)(113,128)(114,134)(115,140)(116,136)(117,142)(118,148)(119,129)(120,135)(121,131)(122,137)(123,143)(124,149)(125,130);
s1 := Sym(150)!(  1,101)(  2,124)(  3,117)(  4,115)(  5,108)(  6,114)(  7,107)(  8,105)(  9,123)( 10,116)( 11,122)( 12,120)( 13,113)( 14,106)( 15,104)( 16,110)( 17,103)( 18,121)( 19,119)( 20,112)( 21,118)( 22,111)( 23,109)( 24,102)( 25,125)( 26, 76)( 27, 99)( 28, 92)( 29, 90)( 30, 83)( 31, 89)( 32, 82)( 33, 80)( 34, 98)( 35, 91)( 36, 97)( 37, 95)( 38, 88)( 39, 81)( 40, 79)( 41, 85)( 42, 78)( 43, 96)( 44, 94)( 45, 87)( 46, 93)( 47, 86)( 48, 84)( 49, 77)( 50,100)( 51,126)( 52,149)( 53,142)( 54,140)( 55,133)( 56,139)( 57,132)( 58,130)( 59,148)( 60,141)( 61,147)( 62,145)( 63,138)( 64,131)( 65,129)( 66,135)( 67,128)( 68,146)( 69,144)( 70,137)( 71,143)( 72,136)( 73,134)( 74,127)( 75,150);
s2 := Sym(150)!(  1,  8)(  2,  7)(  3,  6)(  4, 10)(  5,  9)( 11, 23)( 12, 22)( 13, 21)( 14, 25)( 15, 24)( 16, 18)( 19, 20)( 26, 33)( 27, 32)( 28, 31)( 29, 35)( 30, 34)( 36, 48)( 37, 47)( 38, 46)( 39, 50)( 40, 49)( 41, 43)( 44, 45)( 51, 58)( 52, 57)( 53, 56)( 54, 60)( 55, 59)( 61, 73)( 62, 72)( 63, 71)( 64, 75)( 65, 74)( 66, 68)( 69, 70)( 76, 83)( 77, 82)( 78, 81)( 79, 85)( 80, 84)( 86, 98)( 87, 97)( 88, 96)( 89,100)( 90, 99)( 91, 93)( 94, 95)(101,108)(102,107)(103,106)(104,110)(105,109)(111,123)(112,122)(113,121)(114,125)(115,124)(116,118)(119,120)(126,133)(127,132)(128,131)(129,135)(130,134)(136,148)(137,147)(138,146)(139,150)(140,149)(141,143)(144,145);
poly := sub<Sym(150)|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, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 >;
References : None.
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