Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,6,12}

Atlas Canonical Name {4,6,12}*1152c

Overview

Group
SmallGroup(1152,157559)
Rank
4
Schläfli Type
{4,6,12}
Vertices, edges, …
4, 24, 72, 24
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

3-fold

4-fold

6-fold

8-fold

12-fold

16-fold

24-fold

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

12 facets

4 vertex figures

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

12 facets

4 vertex figures

Representations

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

References

None.

to this polytope.