Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6,6}

Atlas Canonical Name {6,6,6}*1152b

Overview

Group
SmallGroup(1152,157621)
Rank
4
Schläfli Type
{6,6,6}
Vertices, edges, …
16, 48, 48, 6
Order of s0s1s2s3
24
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

3-fold

4-fold

6-fold

12-fold

24-fold

48-fold

72-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)^3> of order 2

6 facets

8 vertex figures

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

6 facets

8 vertex figures

Representations

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

References

None.

to this polytope.