Polytope of Type {4,4}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*800
Also Known As : {4,4}(10,0), {4,4|10}. if this polytope has another name.
Group : SmallGroup(800,1058)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 100, 200, 100
Order of s0s1s2 : 20
Order of s0s1s2s1 : 10
Special Properties :
Toroidal
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Skewing Operation
Facet Of :
{4,4,2} of size 1600
Vertex Figure Of :
{2,4,4} of size 1600
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4}*400
4-fold quotients : {4,4}*200
25-fold quotients : {4,4}*32
50-fold quotients : {2,4}*16, {4,2}*16
100-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4}*1600, {4,8}*1600a, {8,4}*1600a, {4,8}*1600b, {8,4}*1600b
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s0*s1*s0*s1> of order 2.
52 facets:
4 of {2}*4
48 of {4}*8
50 vertex figures:
50 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
50 facets:
50 of {4}*8
50 vertex figures:
50 of {4}*8
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*s2> of order 2.
50 facets:
50 of {4}*8
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
50 facets:
50 of {4}*8
50 vertex figures:
50 of {4}*8
P/N, where N=<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*s2> of order 2.
50 facets:
50 of {4}*8
52 vertex figures:
48 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
25 facets:
25 of {4}*8
26 vertex figures:
24 of {4}*8
2 of {2}*4
P/N, where N=<s0*s1*s0*s1, 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*s2> of order 4.
26 facets:
2 of {2}*4
24 of {4}*8
25 vertex figures:
25 of {4}*8
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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
25 facets:
25 of {4}*8
25 vertex figures:
25 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
20 facets:
20 of {4}*8
20 vertex figures:
20 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
20 facets:
20 of {4}*8
20 vertex figures:
20 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 5.
20 facets:
20 of {4}*8
20 vertex figures:
20 of {4}*8
P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 10.
10 facets:
10 of {4}*8
12 vertex figures:
4 of {2}*4
8 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 10.
10 facets:
10 of {4}*8
12 vertex figures:
8 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 10.
10 facets:
10 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 10.
10 facets:
10 of {4}*8
12 vertex figures:
8 of {4}*8
4 of {2}*4
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 10.
10 facets:
10 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 10.
10 facets:
10 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 10.
10 facets:
10 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 10.
10 facets:
10 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 10.
12 facets:
4 of {2}*4
8 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 10.
12 facets:
4 of {2}*4
8 of {4}*8
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 10.
12 facets:
4 of {2}*4
8 of {4}*8
10 vertex figures:
10 of {4}*8
Permutation Representation (GAP) :
s0 := ( 2, 7)( 3, 13)( 4, 19)( 5, 25)( 6, 21)( 9, 14)( 10, 20)( 11, 16)( 12, 22)( 18, 23)( 27, 32)( 28, 38)( 29, 44)( 30, 50)( 31, 46)( 34, 39)( 35, 45)( 36, 41)( 37, 47)( 43, 48)( 52, 57)( 53, 63)( 54, 69)( 55, 75)( 56, 71)( 59, 64)( 60, 70)( 61, 66)( 62, 72)( 68, 73)( 77, 82)( 78, 88)( 79, 94)( 80,100)( 81, 96)( 84, 89)( 85, 95)( 86, 91)( 87, 97)( 93, 98);;
s1 := ( 2, 19)( 3, 7)( 4, 25)( 5, 13)( 6, 15)( 8, 16)( 10, 22)( 11, 24)( 14, 18)( 17, 21)( 27, 44)( 28, 32)( 29, 50)( 30, 38)( 31, 40)( 33, 41)( 35, 47)( 36, 49)( 39, 43)( 42, 46)( 51, 76)( 52, 94)( 53, 82)( 54,100)( 55, 88)( 56, 90)( 57, 78)( 58, 91)( 59, 84)( 60, 97)( 61, 99)( 62, 87)( 63, 80)( 64, 93)( 65, 81)( 66, 83)( 67, 96)( 68, 89)( 69, 77)( 70, 95)( 71, 92)( 72, 85)( 73, 98)( 74, 86)( 75, 79);;
s2 := ( 1, 58)( 2, 52)( 3, 71)( 4, 70)( 5, 64)( 6, 63)( 7, 57)( 8, 51)( 9, 75)( 10, 69)( 11, 68)( 12, 62)( 13, 56)( 14, 55)( 15, 74)( 16, 73)( 17, 67)( 18, 61)( 19, 60)( 20, 54)( 21, 53)( 22, 72)( 23, 66)( 24, 65)( 25, 59)( 26, 83)( 27, 77)( 28, 96)( 29, 95)( 30, 89)( 31, 88)( 32, 82)( 33, 76)( 34,100)( 35, 94)( 36, 93)( 37, 87)( 38, 81)( 39, 80)( 40, 99)( 41, 98)( 42, 92)( 43, 86)( 44, 85)( 45, 79)( 46, 78)( 47, 97)( 48, 91)( 49, 90)( 50, 84);;
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,
s1*s2*s1*s2*s1*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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(100)!( 2, 7)( 3, 13)( 4, 19)( 5, 25)( 6, 21)( 9, 14)( 10, 20)( 11, 16)( 12, 22)( 18, 23)( 27, 32)( 28, 38)( 29, 44)( 30, 50)( 31, 46)( 34, 39)( 35, 45)( 36, 41)( 37, 47)( 43, 48)( 52, 57)( 53, 63)( 54, 69)( 55, 75)( 56, 71)( 59, 64)( 60, 70)( 61, 66)( 62, 72)( 68, 73)( 77, 82)( 78, 88)( 79, 94)( 80,100)( 81, 96)( 84, 89)( 85, 95)( 86, 91)( 87, 97)( 93, 98);
s1 := Sym(100)!( 2, 19)( 3, 7)( 4, 25)( 5, 13)( 6, 15)( 8, 16)( 10, 22)( 11, 24)( 14, 18)( 17, 21)( 27, 44)( 28, 32)( 29, 50)( 30, 38)( 31, 40)( 33, 41)( 35, 47)( 36, 49)( 39, 43)( 42, 46)( 51, 76)( 52, 94)( 53, 82)( 54,100)( 55, 88)( 56, 90)( 57, 78)( 58, 91)( 59, 84)( 60, 97)( 61, 99)( 62, 87)( 63, 80)( 64, 93)( 65, 81)( 66, 83)( 67, 96)( 68, 89)( 69, 77)( 70, 95)( 71, 92)( 72, 85)( 73, 98)( 74, 86)( 75, 79);
s2 := Sym(100)!( 1, 58)( 2, 52)( 3, 71)( 4, 70)( 5, 64)( 6, 63)( 7, 57)( 8, 51)( 9, 75)( 10, 69)( 11, 68)( 12, 62)( 13, 56)( 14, 55)( 15, 74)( 16, 73)( 17, 67)( 18, 61)( 19, 60)( 20, 54)( 21, 53)( 22, 72)( 23, 66)( 24, 65)( 25, 59)( 26, 83)( 27, 77)( 28, 96)( 29, 95)( 30, 89)( 31, 88)( 32, 82)( 33, 76)( 34,100)( 35, 94)( 36, 93)( 37, 87)( 38, 81)( 39, 80)( 40, 99)( 41, 98)( 42, 92)( 43, 86)( 44, 85)( 45, 79)( 46, 78)( 47, 97)( 48, 91)( 49, 90)( 50, 84);
poly := sub<Sym(100)|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,
s1*s2*s1*s2*s1*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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope
Twisty Puzzle