Polytope of Type {30,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*1080d
if this polytope has a name.
Group : SmallGroup(1080,539)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 90, 270, 18
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {30,6}*360a, {30,6}*360b, {30,6}*360c
   5-fold quotients : {6,6}*216d
   6-fold quotients : {15,6}*180
   9-fold quotients : {10,6}*120, {30,2}*120
   15-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {15,2}*60
   27-fold quotients : {10,2}*40
   30-fold quotients : {3,6}*36, {6,3}*36
   45-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {5,2}*20
   90-fold quotients : {2,3}*12, {3,2}*12
   135-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<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> of order 2.
      12 facets:
         6 of {15}*30
         6 of {30}*60
      45 vertex figures:
         45 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3.
      6 facets:
         6 of {30}*60
      60 vertex figures:
         15 of {6}*12
         45 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      12 facets:
         9 of {10}*20
         3 of {30}*60
      30 vertex figures:
         30 of {6}*12

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