Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,30}

Atlas Canonical Name {6,30}*1620b

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Overview

Group
SmallGroup(1620,136)
Rank
3
Schläfli Type
{6,30}
Vertices, edges, …
27, 405, 135
Order of s0s1s2
45
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

3-fold

5-fold

15-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^2> of order 3

75 facets

9 vertex figures

P/N, where N=<s0*s2*s1*s0*s2*(s1*s0)^2*s2*s1*s2> of order 3

45 facets

9 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle