Polytope of Type {10,4}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4}*2000b
if this polytope has a name.
Group : SmallGroup(2000,919)
Rank : 3
Schlafli Type : {10,4}
Number of vertices, edges, etc : 250, 500, 100
Order of s0s1s2 : 20
Order of s0s1s2s1 : 10
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) :
5-fold quotients : {10,4}*400
10-fold quotients : {10,4}*200
25-fold quotients : {10,4}*80
50-fold quotients : {10,2}*40
100-fold quotients : {5,2}*20
125-fold quotients : {2,4}*16
250-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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 2.
50 facets:
50 of {10}*20
125 vertex figures:
125 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
55 facets:
10 of {5}*10
45 of {10}*20
125 vertex figures:
125 of {4}*8
P/N, where N=<s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 2.
50 facets:
50 of {10}*20
130 vertex figures:
120 of {4}*8
10 of {2}*4
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2> of order 4.
30 facets:
10 of {5}*10
20 of {10}*20
65 vertex figures:
60 of {4}*8
5 of {2}*4
P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s0*s1> of order 5.
40 facets:
25 of {2}*4
15 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2> of order 5.
20 facets:
20 of {10}*20
50 vertex figures:
50 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 10.
15 facets:
10 of {5}*10
5 of {10}*20
25 vertex figures:
25 of {4}*8
P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 10.
10 facets:
10 of {10}*20
30 vertex figures:
20 of {4}*8
10 of {2}*4
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s2*s1*s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2> of order 10.
10 facets:
10 of {10}*20
30 vertex figures:
20 of {4}*8
10 of {2}*4
P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 10.
10 facets:
10 of {10}*20
30 vertex figures:
10 of {2}*4
20 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 20.
8 facets:
6 of {5}*10
2 of {10}*20
15 vertex figures:
10 of {4}*8
5 of {2}*4
P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 25.
4 facets:
4 of {10}*20
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 25.
8 facets:
5 of {2}*4
3 of {10}*20
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1> of order 25.
12 facets:
2 of {10}*20
10 of {2}*4
10 vertex figures:
10 of {4}*8
P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 25.
8 facets:
3 of {10}*20
5 of {2}*4
10 vertex figures:
10 of {4}*8
P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 25.
4 facets:
4 of {10}*20
10 vertex figures:
10 of {4}*8
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 4)( 6, 21)( 7, 25)( 8, 24)( 9, 23)( 10, 22)( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,121)( 32,125)( 33,124)( 34,123)( 35,122)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61, 91)( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 81)( 72, 85)( 73, 84)( 74, 83)( 75, 82);;
s1 := ( 1, 32)( 2, 31)( 3, 35)( 4, 34)( 5, 33)( 6, 7)( 8, 10)( 11,107)( 12,106)( 13,110)( 14,109)( 15,108)( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 57)( 22, 56)( 23, 60)( 24, 59)( 25, 58)( 26, 27)( 28, 30)( 36,102)( 37,101)( 38,105)( 39,104)( 40,103)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 78)( 46, 52)( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)( 66, 97)( 67, 96)( 68,100)( 69, 99)( 70, 98)( 71, 72)( 73, 75)( 86,117)( 87,116)( 88,120)( 89,119)( 90,118)( 91, 92)( 93, 95)(111,112)(113,115);;
s2 := ( 6, 76)( 7, 77)( 8, 78)( 9, 79)( 10, 80)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 16,101)( 17,102)( 18,103)( 19,104)( 20,105)( 21, 51)( 22, 52)( 23, 53)( 24, 54)( 25, 55)( 31, 86)( 32, 87)( 33, 88)( 34, 89)( 35, 90)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46, 61)( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 56, 96)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(125)!( 2, 5)( 3, 4)( 6, 21)( 7, 25)( 8, 24)( 9, 23)( 10, 22)( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,121)( 32,125)( 33,124)( 34,123)( 35,122)( 36,116)( 37,120)( 38,119)( 39,118)( 40,117)( 41,111)( 42,115)( 43,114)( 44,113)( 45,112)( 46,106)( 47,110)( 48,109)( 49,108)( 50,107)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61, 91)( 62, 95)( 63, 94)( 64, 93)( 65, 92)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 81)( 72, 85)( 73, 84)( 74, 83)( 75, 82);
s1 := Sym(125)!( 1, 32)( 2, 31)( 3, 35)( 4, 34)( 5, 33)( 6, 7)( 8, 10)( 11,107)( 12,106)( 13,110)( 14,109)( 15,108)( 16, 82)( 17, 81)( 18, 85)( 19, 84)( 20, 83)( 21, 57)( 22, 56)( 23, 60)( 24, 59)( 25, 58)( 26, 27)( 28, 30)( 36,102)( 37,101)( 38,105)( 39,104)( 40,103)( 41, 77)( 42, 76)( 43, 80)( 44, 79)( 45, 78)( 46, 52)( 47, 51)( 48, 55)( 49, 54)( 50, 53)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)( 66, 97)( 67, 96)( 68,100)( 69, 99)( 70, 98)( 71, 72)( 73, 75)( 86,117)( 87,116)( 88,120)( 89,119)( 90,118)( 91, 92)( 93, 95)(111,112)(113,115);
s2 := Sym(125)!( 6, 76)( 7, 77)( 8, 78)( 9, 79)( 10, 80)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 16,101)( 17,102)( 18,103)( 19,104)( 20,105)( 21, 51)( 22, 52)( 23, 53)( 24, 54)( 25, 55)( 31, 86)( 32, 87)( 33, 88)( 34, 89)( 35, 90)( 41,111)( 42,112)( 43,113)( 44,114)( 45,115)( 46, 61)( 47, 62)( 48, 63)( 49, 64)( 50, 65)( 56, 96)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110);
poly := sub<Sym(125)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope
Twisty Puzzle