Polytope of Type {3,4,4,6} with group SmallGroup(1152,157559)

Overview

Group: SmallGroup(1152,157559)
Rank: 5
Schläfli Type: {3,4,4,6}
Vertices, edges, …: 6, 12, 16, 12, 6
Order of s₀s₁s₂s₃s₄: 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁: 2
Also known as: {{3,4},{4,4|2},{4,6|2}}. if this polytope has another name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{3,4,2,6}*576

3-fold

{3,4,4,2}*384b

4-fold

6-fold

{3,4,2,2}*192

8-fold

12-fold

16-fold

{3,2,2,3}*72

24-fold

{3,2,2,2}*48

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 2

6 facets

6 of 2-fold non-regular quotient of {3,4,4}*192b

4 vertex figures

2 of {4,4,6}*192
2 of {2,4,6}*96a

Representations

Permutation Representation (GAP)

s0 := (  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143);;
s1 := (  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 17)(  6, 20)(  7, 19)(  8, 18)(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)(111,123)(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)(119,131)(120,130)(134,136)(138,140)(142,144);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73,110)( 74,109)( 75,112)( 76,111)( 77,114)( 78,113)( 79,116)( 80,115)( 81,118)( 82,117)( 83,120)( 84,119)( 85,122)( 86,121)( 87,124)( 88,123)( 89,126)( 90,125)( 91,128)( 92,127)( 93,130)( 94,129)( 95,132)( 96,131)( 97,134)( 98,133)( 99,136)(100,135)(101,138)(102,137)(103,140)(104,139)(105,142)(106,141)(107,144)(108,143);;
s3 := (  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 81)(  6, 82)(  7, 83)(  8, 84)(  9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 85)( 14, 86)( 15, 87)( 16, 88)( 17, 93)( 18, 94)( 19, 95)( 20, 96)( 21, 89)( 22, 90)( 23, 91)( 24, 92)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,105)( 30,106)( 31,107)( 32,108)( 33,101)( 34,102)( 35,103)( 36,104)( 37,109)( 38,110)( 39,111)( 40,112)( 41,117)( 42,118)( 43,119)( 44,120)( 45,113)( 46,114)( 47,115)( 48,116)( 49,121)( 50,122)( 51,123)( 52,124)( 53,129)( 54,130)( 55,131)( 56,132)( 57,125)( 58,126)( 59,127)( 60,128)( 61,133)( 62,134)( 63,135)( 64,136)( 65,141)( 66,142)( 67,143)( 68,144)( 69,137)( 70,138)( 71,139)( 72,140);;
s4 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 97,101)( 98,102)( 99,103)(100,104)(109,113)(110,114)(111,115)(112,116)(121,125)(122,126)(123,127)(124,128)(133,137)(134,138)(135,139)(136,140);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(144)!(  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 39, 40)( 43, 44)( 47, 48)( 49, 61)( 50, 62)( 51, 64)( 52, 63)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 69)( 58, 70)( 59, 72)( 60, 71)( 75, 76)( 79, 80)( 83, 84)( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)( 94,106)( 95,108)( 96,107)(111,112)(115,116)(119,120)(121,133)(122,134)(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)(131,144)(132,143);
s1 := Sym(144)!(  1, 13)(  2, 16)(  3, 15)(  4, 14)(  5, 17)(  6, 20)(  7, 19)(  8, 18)(  9, 21)( 10, 24)( 11, 23)( 12, 22)( 26, 28)( 30, 32)( 34, 36)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 85)( 74, 88)( 75, 87)( 76, 86)( 77, 89)( 78, 92)( 79, 91)( 80, 90)( 81, 93)( 82, 96)( 83, 95)( 84, 94)( 98,100)(102,104)(106,108)(109,121)(110,124)(111,123)(112,122)(113,125)(114,128)(115,127)(116,126)(117,129)(118,132)(119,131)(120,130)(134,136)(138,140)(142,144);
s2 := Sym(144)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73,110)( 74,109)( 75,112)( 76,111)( 77,114)( 78,113)( 79,116)( 80,115)( 81,118)( 82,117)( 83,120)( 84,119)( 85,122)( 86,121)( 87,124)( 88,123)( 89,126)( 90,125)( 91,128)( 92,127)( 93,130)( 94,129)( 95,132)( 96,131)( 97,134)( 98,133)( 99,136)(100,135)(101,138)(102,137)(103,140)(104,139)(105,142)(106,141)(107,144)(108,143);
s3 := Sym(144)!(  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 81)(  6, 82)(  7, 83)(  8, 84)(  9, 77)( 10, 78)( 11, 79)( 12, 80)( 13, 85)( 14, 86)( 15, 87)( 16, 88)( 17, 93)( 18, 94)( 19, 95)( 20, 96)( 21, 89)( 22, 90)( 23, 91)( 24, 92)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,105)( 30,106)( 31,107)( 32,108)( 33,101)( 34,102)( 35,103)( 36,104)( 37,109)( 38,110)( 39,111)( 40,112)( 41,117)( 42,118)( 43,119)( 44,120)( 45,113)( 46,114)( 47,115)( 48,116)( 49,121)( 50,122)( 51,123)( 52,124)( 53,129)( 54,130)( 55,131)( 56,132)( 57,125)( 58,126)( 59,127)( 60,128)( 61,133)( 62,134)( 63,135)( 64,136)( 65,141)( 66,142)( 67,143)( 68,144)( 69,137)( 70,138)( 71,139)( 72,140);
s4 := Sym(144)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)( 13, 17)( 14, 18)( 15, 19)( 16, 20)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 73, 77)( 74, 78)( 75, 79)( 76, 80)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 97,101)( 98,102)( 99,103)(100,104)(109,113)(110,114)(111,115)(112,116)(121,125)(122,126)(123,127)(124,128)(133,137)(134,138)(135,139)(136,140);
poly := sub<Sym(144)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

References

None.

to this polytope.