Polytope of Type {15,6,3} with group SmallGroup(1620,135)

Overview

Group: SmallGroup(1620,135)
Rank: 4
Schläfli Type: {15,6,3}
Vertices, edges, …: 15, 135, 27, 9
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

{15,6,3}*540

5-fold

{3,6,3}*324a

9-fold

{15,2,3}*180

15-fold

{3,6,3}*108

27-fold

{5,2,3}*60

45-fold

{3,2,3}*36

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=<(s2*s1*s2*s3)^2> of order 3

5 facets

2 of {15,6}*180
3 of {15,2}*60

15 vertex figures

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

Representations

Permutation Representation (GAP)

s0 := (  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)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64, 73)( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(100,127)(101,129)(102,128)(103,133)(104,135)(105,134)(106,130)(107,132)(108,131)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122);;
s1 := (  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, 59)( 47, 58)( 48, 60)( 49, 56)( 50, 55)( 51, 57)( 52, 62)( 53, 61)( 54, 63)( 64, 86)( 65, 85)( 66, 87)( 67, 83)( 68, 82)( 69, 84)( 70, 89)( 71, 88)( 72, 90)( 73, 77)( 74, 76)( 75, 78)( 79, 80)( 91,107)( 92,106)( 93,108)( 94,104)( 95,103)( 96,105)( 97,101)( 98,100)( 99,102)(109,134)(110,133)(111,135)(112,131)(113,130)(114,132)(115,128)(116,127)(117,129)(118,125)(119,124)(120,126)(121,122);;
s2 := (  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);;
s3 := (  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);;
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, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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,  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)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64, 73)( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(100,127)(101,129)(102,128)(103,133)(104,135)(105,134)(106,130)(107,132)(108,131)(109,118)(110,120)(111,119)(112,124)(113,126)(114,125)(115,121)(116,123)(117,122);
s1 := 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, 59)( 47, 58)( 48, 60)( 49, 56)( 50, 55)( 51, 57)( 52, 62)( 53, 61)( 54, 63)( 64, 86)( 65, 85)( 66, 87)( 67, 83)( 68, 82)( 69, 84)( 70, 89)( 71, 88)( 72, 90)( 73, 77)( 74, 76)( 75, 78)( 79, 80)( 91,107)( 92,106)( 93,108)( 94,104)( 95,103)( 96,105)( 97,101)( 98,100)( 99,102)(109,134)(110,133)(111,135)(112,131)(113,130)(114,132)(115,128)(116,127)(117,129)(118,125)(119,124)(120,126)(121,122);
s2 := 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);
s3 := 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);
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, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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.

to this polytope.