Polytope of Type {12,4}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,32552)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 144, 288, 48
Order of s0s1s2 : 8
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,4}*576
4-fold quotients : {12,4}*288
8-fold quotients : {6,4}*144
9-fold quotients : {4,4}*128
16-fold quotients : {6,4}*72
18-fold quotients : {4,4}*64
36-fold quotients : {4,4}*32
72-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 2.
30 facets:
18 of {12}*24
12 of {6}*12
72 vertex figures:
72 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
24 facets:
24 of {12}*24
72 vertex figures:
72 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 2.
24 facets:
24 of {12}*24
72 vertex figures:
72 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 2.
24 facets:
24 of {12}*24
72 vertex figures:
72 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 3.
16 facets:
16 of {12}*24
48 vertex figures:
48 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
32 facets:
24 of {4}*8
8 of {12}*24
48 vertex figures:
48 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 4.
18 facets:
12 of {6}*12
6 of {12}*24
36 vertex figures:
36 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 4.
15 facets:
9 of {12}*24
6 of {6}*12
36 vertex figures:
36 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s2> of order 4.
16 facets:
9 of {12}*24
2 of {3}*6
5 of {6}*12
36 vertex figures:
36 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1> of order 4.
12 facets:
12 of {12}*24
36 vertex figures:
36 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
12 facets:
12 of {12}*24
36 vertex figures:
36 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 4.
12 facets:
12 of {12}*24
36 vertex figures:
36 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 4.
12 facets:
12 of {12}*24
36 vertex figures:
36 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
16 facets:
12 of {4}*8
4 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 6.
8 facets:
8 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0> of order 6.
8 facets:
8 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 6.
8 facets:
8 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 6.
8 facets:
8 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
16 facets:
12 of {4}*8
4 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 6.
18 facets:
12 of {4}*8
4 of {6}*12
2 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 6.
8 facets:
8 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 6.
10 facets:
6 of {12}*24
4 of {6}*12
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 6.
8 facets:
8 of {12}*24
24 vertex figures:
24 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s2> of order 6.
22 facets:
4 of {12}*24
12 of {2}*4
6 of {4}*8
24 vertex figures:
24 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 12.
6 facets:
4 of {6}*12
2 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 12.
6 facets:
2 of {12}*24
4 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 12.
12 facets:
6 of {2}*4
3 of {4}*8
2 of {6}*12
1 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 12.
10 facets:
6 of {4}*8
2 of {3}*6
1 of {12}*24
1 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 12.
4 facets:
4 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 12.
4 facets:
4 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 12.
4 facets:
4 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 12.
4 facets:
4 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 12.
8 facets:
6 of {4}*8
2 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s1, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 12.
4 facets:
4 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 12.
4 facets:
4 of {12}*24
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 12.
11 facets:
2 of {12}*24
6 of {2}*4
3 of {4}*8
12 vertex figures:
12 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 12.
5 facets:
3 of {12}*24
2 of {6}*12
12 vertex figures:
12 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 12.
9 facets:
6 of {4}*8
2 of {6}*12
1 of {12}*24
12 vertex figures:
12 of {4}*8
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 37, 64)( 38, 66)( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 55)( 47, 57)( 48, 56)( 49, 61)( 50, 63)( 51, 62)( 52, 58)( 53, 60)( 54, 59)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)(117,140)(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)(125,132)(126,131);;
s1 := ( 1, 40)( 2, 38)( 3, 45)( 4, 37)( 5, 44)( 6, 42)( 7, 43)( 8, 41)( 9, 39)( 10, 49)( 11, 47)( 12, 54)( 13, 46)( 14, 53)( 15, 51)( 16, 52)( 17, 50)( 18, 48)( 19, 67)( 20, 65)( 21, 72)( 22, 64)( 23, 71)( 24, 69)( 25, 70)( 26, 68)( 27, 66)( 28, 58)( 29, 56)( 30, 63)( 31, 55)( 32, 62)( 33, 60)( 34, 61)( 35, 59)( 36, 57)( 73, 76)( 75, 81)( 77, 80)( 82, 85)( 84, 90)( 86, 89)( 91,103)( 92,101)( 93,108)( 94,100)( 95,107)( 96,105)( 97,106)( 98,104)( 99,102)(109,112)(111,117)(113,116)(118,121)(120,126)(122,125)(127,139)(128,137)(129,144)(130,136)(131,143)(132,141)(133,142)(134,140)(135,138);;
s2 := ( 1, 73)( 2, 75)( 3, 74)( 4, 77)( 5, 76)( 6, 78)( 7, 81)( 8, 80)( 9, 79)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)( 17, 89)( 18, 88)( 19, 91)( 20, 93)( 21, 92)( 22, 95)( 23, 94)( 24, 96)( 25, 99)( 26, 98)( 27, 97)( 28,100)( 29,102)( 30,101)( 31,104)( 32,103)( 33,105)( 34,108)( 35,107)( 36,106)( 37,109)( 38,111)( 39,110)( 40,113)( 41,112)( 42,114)( 43,117)( 44,116)( 45,115)( 46,118)( 47,120)( 48,119)( 49,122)( 50,121)( 51,123)( 52,126)( 53,125)( 54,124)( 55,127)( 56,129)( 57,128)( 58,131)( 59,130)( 60,132)( 61,135)( 62,134)( 63,133)( 64,136)( 65,138)( 66,137)( 67,140)( 68,139)( 69,141)( 70,144)( 71,143)( 72,142);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*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(144)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 37, 64)( 38, 66)( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 55)( 47, 57)( 48, 56)( 49, 61)( 50, 63)( 51, 62)( 52, 58)( 53, 60)( 54, 59)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)(114,143)(115,139)(116,141)(117,140)(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)(125,132)(126,131);
s1 := Sym(144)!( 1, 40)( 2, 38)( 3, 45)( 4, 37)( 5, 44)( 6, 42)( 7, 43)( 8, 41)( 9, 39)( 10, 49)( 11, 47)( 12, 54)( 13, 46)( 14, 53)( 15, 51)( 16, 52)( 17, 50)( 18, 48)( 19, 67)( 20, 65)( 21, 72)( 22, 64)( 23, 71)( 24, 69)( 25, 70)( 26, 68)( 27, 66)( 28, 58)( 29, 56)( 30, 63)( 31, 55)( 32, 62)( 33, 60)( 34, 61)( 35, 59)( 36, 57)( 73, 76)( 75, 81)( 77, 80)( 82, 85)( 84, 90)( 86, 89)( 91,103)( 92,101)( 93,108)( 94,100)( 95,107)( 96,105)( 97,106)( 98,104)( 99,102)(109,112)(111,117)(113,116)(118,121)(120,126)(122,125)(127,139)(128,137)(129,144)(130,136)(131,143)(132,141)(133,142)(134,140)(135,138);
s2 := Sym(144)!( 1, 73)( 2, 75)( 3, 74)( 4, 77)( 5, 76)( 6, 78)( 7, 81)( 8, 80)( 9, 79)( 10, 82)( 11, 84)( 12, 83)( 13, 86)( 14, 85)( 15, 87)( 16, 90)( 17, 89)( 18, 88)( 19, 91)( 20, 93)( 21, 92)( 22, 95)( 23, 94)( 24, 96)( 25, 99)( 26, 98)( 27, 97)( 28,100)( 29,102)( 30,101)( 31,104)( 32,103)( 33,105)( 34,108)( 35,107)( 36,106)( 37,109)( 38,111)( 39,110)( 40,113)( 41,112)( 42,114)( 43,117)( 44,116)( 45,115)( 46,118)( 47,120)( 48,119)( 49,122)( 50,121)( 51,123)( 52,126)( 53,125)( 54,124)( 55,127)( 56,129)( 57,128)( 58,131)( 59,130)( 60,132)( 61,135)( 62,134)( 63,133)( 64,136)( 65,138)( 66,137)( 67,140)( 68,139)( 69,141)( 70,144)( 71,143)( 72,142);
poly := sub<Sym(144)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*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
Twisty Puzzle