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