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