Polytope of Type {4,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30394)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 4, 108, 108, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,6,4}*864a
3-fold quotients : {4,6,4}*576b
4-fold quotients : {2,6,4}*432
6-fold quotients : {2,6,4}*288
12-fold quotients : {2,6,4}*144
27-fold quotients : {4,2,4}*64
54-fold quotients : {2,2,4}*32, {4,2,2}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s3> of order 2.
18 facets:
18 of {4,6}*48a
4 vertex figures:
4 of 2-fold non-regular quotient of {6,4}*432a
P/N, where N=<s1*s3*s2*s1*s2*s1*s3*s2*s1*s2> of order 3.
12 facets:
12 of {4,6}*48a
4 vertex figures:
4 of 3-fold non-regular quotient of {6,4}*432a
P/N, where N=<s2*s3*s2*s1*s3*s2*s1*s3*s2*s3> of order 3.
16 facets:
10 of {4,6}*48a
6 of {4,2}*16
4 vertex figures:
4 of 3-fold non-regular quotient of {6,4}*432a
P/N, where N=<s1*s2*s1*s2, s3*s2*s1*s2*s3*s2*s1*s3*s2*s3> of order 6.
8 facets:
3 of {4,2}*16
5 of {4,6}*48a
4 vertex figures:
4 of 6-fold non-regular quotient of {6,4}*432a
P/N, where N=<s2*s3*s2*s3, s1*s2*s3*s2*s1*s3> of order 6.
6 facets:
6 of {4,6}*48a
4 vertex figures:
4 of 6-fold non-regular quotient of {6,4}*432a
Permutation Representation (GAP) :
s0 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)( 79,106)( 80,107)( 81,108);;
s1 := ( 1, 55)( 2, 56)( 3, 57)( 4, 63)( 5, 61)( 6, 62)( 7, 59)( 8, 60)( 9, 58)( 10, 75)( 11, 73)( 12, 74)( 13, 80)( 14, 81)( 15, 79)( 16, 76)( 17, 77)( 18, 78)( 19, 65)( 20, 66)( 21, 64)( 22, 70)( 23, 71)( 24, 72)( 25, 69)( 26, 67)( 27, 68)( 28, 82)( 29, 83)( 30, 84)( 31, 90)( 32, 88)( 33, 89)( 34, 86)( 35, 87)( 36, 85)( 37,102)( 38,100)( 39,101)( 40,107)( 41,108)( 42,106)( 43,103)( 44,104)( 45,105)( 46, 92)( 47, 93)( 48, 91)( 49, 97)( 50, 98)( 51, 99)( 52, 96)( 53, 94)( 54, 95);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 5)( 7, 26)( 8, 25)( 9, 27)( 13, 23)( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 31, 32)( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)( 47, 48)( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)( 67, 77)( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)( 85, 86)( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)(101,102);;
s3 := ( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 59)( 61, 63)( 64, 79)( 65, 81)( 66, 80)( 67, 74)( 68, 73)( 69, 75)( 70, 78)( 71, 77)( 72, 76)( 83, 84)( 85, 86)( 88, 90)( 91,106)( 92,108)( 93,107)( 94,101)( 95,100)( 96,102)( 97,105)( 98,104)( 99,103);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(108)!( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)( 79,106)( 80,107)( 81,108);
s1 := Sym(108)!( 1, 55)( 2, 56)( 3, 57)( 4, 63)( 5, 61)( 6, 62)( 7, 59)( 8, 60)( 9, 58)( 10, 75)( 11, 73)( 12, 74)( 13, 80)( 14, 81)( 15, 79)( 16, 76)( 17, 77)( 18, 78)( 19, 65)( 20, 66)( 21, 64)( 22, 70)( 23, 71)( 24, 72)( 25, 69)( 26, 67)( 27, 68)( 28, 82)( 29, 83)( 30, 84)( 31, 90)( 32, 88)( 33, 89)( 34, 86)( 35, 87)( 36, 85)( 37,102)( 38,100)( 39,101)( 40,107)( 41,108)( 42,106)( 43,103)( 44,104)( 45,105)( 46, 92)( 47, 93)( 48, 91)( 49, 97)( 50, 98)( 51, 99)( 52, 96)( 53, 94)( 54, 95);
s2 := Sym(108)!( 1, 10)( 2, 12)( 3, 11)( 4, 5)( 7, 26)( 8, 25)( 9, 27)( 13, 23)( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 31, 32)( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)( 47, 48)( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)( 67, 77)( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)( 85, 86)( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)(101,102);
s3 := Sym(108)!( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)( 56, 57)( 58, 59)( 61, 63)( 64, 79)( 65, 81)( 66, 80)( 67, 74)( 68, 73)( 69, 75)( 70, 78)( 71, 77)( 72, 76)( 83, 84)( 85, 86)( 88, 90)( 91,106)( 92,108)( 93,107)( 94,101)( 95,100)( 96,102)( 97,105)( 98,104)( 99,103);
poly := sub<Sym(108)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;
References : None.
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