Polytope of Type {4,6,3} with group SmallGroup(576,8355)

Overview

Group: SmallGroup(576,8355)
Rank: 4
Schläfli Type: {4,6,3}
Vertices, edges, …: 4, 48, 36, 12
Order of s₀s₁s₂s₃: 12
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{2,6,3}*288

3-fold

{4,6,3}*192

4-fold

{4,6,3}*144

6-fold

{2,6,3}*96

8-fold

{2,6,3}*72

12-fold

24-fold

{2,2,3}*24

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2)^2*s1*s3*s2*s1*s2*s3> of order 2

6 facets

6 of {4,6}*48a

4 vertex figures

4 of 2-fold non-regular quotient of {6,3}*144

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

6 facets

3 of {4,2}*16
3 of {4,6}*48a

4 vertex figures

4 of 3-fold non-regular quotient of {6,3}*144

Representations

Permutation Representation (GAP)

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

Permutation Representation (Magma)

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

References

None.

to this polytope.