Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,15}

Atlas Canonical Name {3,6,15}*1620a

Overview

Group
SmallGroup(1620,135)
Rank
4
Schläfli Type
{3,6,15}
Vertices, edges, …
9, 27, 135, 15
Order of s0s1s2s3
45
Order of s0s1s2s3s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

5-fold

9-fold

15-fold

27-fold

45-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*s0*s2)^2*s1> of order 3

15 facets

  • 15 of 3-fold non-regular quotient of {3,6}*108

5 vertex figures

Representations

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

References

None.

to this polytope.