Polytope of Type {6,3}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3}*1728
Also Known As : {6,3}(12,0), {6,3}24. if this polytope has another name.
Group : SmallGroup(1728,12317)
Rank : 3
Schlafli Type : {6,3}
Number of vertices, edges, etc : 288, 432, 144
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Toroidal
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,3}*576
4-fold quotients : {6,3}*432
9-fold quotients : {6,3}*192
12-fold quotients : {6,3}*144
16-fold quotients : {6,3}*108
36-fold quotients : {6,3}*48
48-fold quotients : {6,3}*36
72-fold quotients : {3,3}*24
144-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
72 facets:
72 of {6}*12
144 vertex figures:
144 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1> of order 2.
74 facets:
4 of {3}*6
70 of {6}*12
144 vertex figures:
144 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 3.
48 facets:
48 of {6}*12
96 vertex figures:
96 of {3}*6
P/N, where N=<s0*s1*s0*s1> of order 3.
50 facets:
3 of {2}*4
47 of {6}*12
96 vertex figures:
96 of {3}*6
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 4.
36 facets:
36 of {6}*12
72 vertex figures:
72 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
38 facets:
4 of {3}*6
34 of {6}*12
72 vertex figures:
72 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 4.
36 facets:
36 of {6}*12
72 vertex figures:
72 of {3}*6
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
36 facets:
36 of {6}*12
72 vertex figures:
72 of {3}*6
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 4.
36 facets:
36 of {6}*12
72 vertex figures:
72 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 6.
26 facets:
4 of {3}*6
22 of {6}*12
48 vertex figures:
48 of {3}*6
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 6.
24 facets:
24 of {6}*12
48 vertex figures:
48 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 6.
24 facets:
24 of {6}*12
48 vertex figures:
48 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 8.
18 facets:
18 of {6}*12
36 vertex figures:
36 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 8.
18 facets:
18 of {6}*12
36 vertex figures:
36 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0> of order 8.
18 facets:
18 of {6}*12
36 vertex figures:
36 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
19 facets:
2 of {3}*6
17 of {6}*12
36 vertex figures:
36 of {3}*6
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1> of order 8.
18 facets:
18 of {6}*12
36 vertex figures:
36 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 12.
12 facets:
12 of {6}*12
24 vertex figures:
24 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 12.
12 facets:
12 of {6}*12
24 vertex figures:
24 of {3}*6
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 12.
12 facets:
12 of {6}*12
24 vertex figures:
24 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s2> of order 12.
14 facets:
4 of {3}*6
10 of {6}*12
24 vertex figures:
24 of {3}*6
P/N, where N=<s0*s1*s0*s1*s0*s1, s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2> of order 12.
14 facets:
4 of {3}*6
10 of {6}*12
24 vertex figures:
24 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 12.
12 facets:
12 of {6}*12
24 vertex figures:
24 of {3}*6
P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1> of order 24.
6 facets:
6 of {6}*12
12 vertex figures:
12 of {3}*6
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 9, 14)( 10, 13)( 11, 15)( 12, 16)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)( 51, 52)( 53, 54)( 57, 62)( 58, 61)( 59, 63)( 60, 64)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 94)( 74, 93)( 75, 95)( 76, 96)( 77, 90)( 78, 89)( 79, 91)( 80, 92)( 99,100)(101,102)(105,110)(106,109)(107,111)(108,112)(113,129)(114,130)(115,132)(116,131)(117,134)(118,133)(119,135)(120,136)(121,142)(122,141)(123,143)(124,144)(125,138)(126,137)(127,139)(128,140);;
s1 := ( 2, 4)( 5, 13)( 6, 16)( 7, 15)( 8, 14)( 9, 11)( 18, 20)( 21, 29)( 22, 32)( 23, 31)( 24, 30)( 25, 27)( 34, 36)( 37, 45)( 38, 48)( 39, 47)( 40, 46)( 41, 43)( 49,129)( 50,132)( 51,131)( 52,130)( 53,141)( 54,144)( 55,143)( 56,142)( 57,139)( 58,138)( 59,137)( 60,140)( 61,133)( 62,136)( 63,135)( 64,134)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69,109)( 70,112)( 71,111)( 72,110)( 73,107)( 74,106)( 75,105)( 76,108)( 77,101)( 78,104)( 79,103)( 80,102)( 81,113)( 82,116)( 83,115)( 84,114)( 85,125)( 86,128)( 87,127)( 88,126)( 89,123)( 90,122)( 91,121)( 92,124)( 93,117)( 94,120)( 95,119)( 96,118);;
s2 := ( 1, 55)( 2, 56)( 3, 54)( 4, 53)( 5, 52)( 6, 51)( 7, 49)( 8, 50)( 9, 58)( 10, 57)( 11, 59)( 12, 60)( 13, 62)( 14, 61)( 15, 63)( 16, 64)( 17, 71)( 18, 72)( 19, 70)( 20, 69)( 21, 68)( 22, 67)( 23, 65)( 24, 66)( 25, 74)( 26, 73)( 27, 75)( 28, 76)( 29, 78)( 30, 77)( 31, 79)( 32, 80)( 33, 87)( 34, 88)( 35, 86)( 36, 85)( 37, 84)( 38, 83)( 39, 81)( 40, 82)( 41, 90)( 42, 89)( 43, 91)( 44, 92)( 45, 94)( 46, 93)( 47, 95)( 48, 96)( 97,103)( 98,104)( 99,102)(100,101)(105,106)(109,110)(113,119)(114,120)(115,118)(116,117)(121,122)(125,126)(129,135)(130,136)(131,134)(132,133)(137,138)(141,142);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(144)!( 3, 4)( 5, 6)( 9, 14)( 10, 13)( 11, 15)( 12, 16)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)( 51, 52)( 53, 54)( 57, 62)( 58, 61)( 59, 63)( 60, 64)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 94)( 74, 93)( 75, 95)( 76, 96)( 77, 90)( 78, 89)( 79, 91)( 80, 92)( 99,100)(101,102)(105,110)(106,109)(107,111)(108,112)(113,129)(114,130)(115,132)(116,131)(117,134)(118,133)(119,135)(120,136)(121,142)(122,141)(123,143)(124,144)(125,138)(126,137)(127,139)(128,140);
s1 := Sym(144)!( 2, 4)( 5, 13)( 6, 16)( 7, 15)( 8, 14)( 9, 11)( 18, 20)( 21, 29)( 22, 32)( 23, 31)( 24, 30)( 25, 27)( 34, 36)( 37, 45)( 38, 48)( 39, 47)( 40, 46)( 41, 43)( 49,129)( 50,132)( 51,131)( 52,130)( 53,141)( 54,144)( 55,143)( 56,142)( 57,139)( 58,138)( 59,137)( 60,140)( 61,133)( 62,136)( 63,135)( 64,134)( 65, 97)( 66,100)( 67, 99)( 68, 98)( 69,109)( 70,112)( 71,111)( 72,110)( 73,107)( 74,106)( 75,105)( 76,108)( 77,101)( 78,104)( 79,103)( 80,102)( 81,113)( 82,116)( 83,115)( 84,114)( 85,125)( 86,128)( 87,127)( 88,126)( 89,123)( 90,122)( 91,121)( 92,124)( 93,117)( 94,120)( 95,119)( 96,118);
s2 := Sym(144)!( 1, 55)( 2, 56)( 3, 54)( 4, 53)( 5, 52)( 6, 51)( 7, 49)( 8, 50)( 9, 58)( 10, 57)( 11, 59)( 12, 60)( 13, 62)( 14, 61)( 15, 63)( 16, 64)( 17, 71)( 18, 72)( 19, 70)( 20, 69)( 21, 68)( 22, 67)( 23, 65)( 24, 66)( 25, 74)( 26, 73)( 27, 75)( 28, 76)( 29, 78)( 30, 77)( 31, 79)( 32, 80)( 33, 87)( 34, 88)( 35, 86)( 36, 85)( 37, 84)( 38, 83)( 39, 81)( 40, 82)( 41, 90)( 42, 89)( 43, 91)( 44, 92)( 45, 94)( 46, 93)( 47, 95)( 48, 96)( 97,103)( 98,104)( 99,102)(100,101)(105,106)(109,110)(113,119)(114,120)(115,118)(116,117)(121,122)(125,126)(129,135)(130,136)(131,134)(132,133)(137,138)(141,142);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope
Twisty Puzzle