Overview
- Group
- SmallGroup(1728,46672)
- Rank
- 4
- Schläfli Type
- {4,6,4}
- Vertices, edges, …
- 4, 108, 108, 36
- Order of s0s1s2s3
- 12
- Order of s0s1s2s3s2s1
- 2
- Also known as
- if this polytope has a name.
Special Properties
- Universal
- Orientable
- Flat
Quotients maximal quotients in bold
2-fold
3-fold
6-fold
9-fold
12-fold
18-fold
27-fold
36-fold
54-fold
72-fold
108-fold
Covers minimal covers in bold
None in this atlas.
Irregular Quotients of which this is a minimal cover
Click an entry to reveal its facets and vertex figures.
P/N, where N=<((s2*s1)^2*s2*s3)^2> of order 2
18 facets
- 18 of {4,6}*48a
4 vertex figures
- 4 of 2-fold non-regular quotient of {6,4}*432b
P/N, where N=<s1*s3*s2*s1*s3*(s2*s1)^2*s3*s2*s3> of order 3
12 facets
- 12 of {4,6}*48a
4 vertex figures
- 4 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s1*s2)^2> of order 3
18 facets
4 vertex figures
- 4 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s2*s1*s2*s3)^2> of order 3
12 facets
- 12 of {4,6}*48a
4 vertex figures
- 4 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s1*s2)^2*(s3*s2*s1)^2*(s2*s3)^2> of order 3
12 facets
- 12 of {4,6}*48a
4 vertex figures
- 4 of 3-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s2*s1*s2*s3)^2, ((s2*s1)^2*s2*s3)^2> of order 6
6 facets
- 6 of {4,6}*48a
4 vertex figures
- 4 of 6-fold non-regular quotient of {6,4}*432b
P/N, where N=<s3*s2*s1*s2*s3*s2*s1*s3*s2*s3, s1*s2*s3*(s2*s1)^2*s2*s3*s2*s1*s2> of order 6
6 facets
- 6 of {4,6}*48a
4 vertex figures
- 4 of 6-fold non-regular quotient of {6,4}*432b
P/N, where N=<(s1*s2)^2, s1*s3*s2*s1*s2*s3> of order 9
8 facets
4 vertex figures
- 4 of 9-fold non-regular quotient of {6,4}*432b
P/N, where N=<s2*s1*s3*s2*s1*s2*s3*s2, (s1*s2)^2*s1*s3*s2*s1*s2*s3> of order 9
6 facets
4 vertex figures
- 4 of 9-fold non-regular quotient of {6,4}*432b
Representations
Permutation Representation (GAP)
s0 := ( 1, 55)( 2, 56)( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108);; s1 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)( 44, 51)( 45, 50)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64,100)( 65,102)( 66,101)( 67,106)( 68,108)( 69,107)( 70,103)( 71,105)( 72,104)( 73, 91)( 74, 93)( 75, 92)( 76, 97)( 77, 99)( 78, 98)( 79, 94)( 80, 96)( 81, 95);; s2 := ( 1, 11)( 2, 10)( 3, 12)( 4, 15)( 5, 14)( 6, 13)( 7, 16)( 8, 18)( 9, 17)( 19, 20)( 22, 24)( 26, 27)( 28, 38)( 29, 37)( 30, 39)( 31, 42)( 32, 41)( 33, 40)( 34, 43)( 35, 45)( 36, 44)( 46, 47)( 49, 51)( 53, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 69)( 59, 68)( 60, 67)( 61, 70)( 62, 72)( 63, 71)( 73, 74)( 76, 78)( 80, 81)( 82, 92)( 83, 91)( 84, 93)( 85, 96)( 86, 95)( 87, 94)( 88, 97)( 89, 99)( 90, 98)(100,101)(103,105)(107,108);; s3 := ( 2, 8)( 3, 6)( 4, 7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)( 48, 51)( 49, 52)( 56, 62)( 57, 60)( 58, 61)( 65, 71)( 66, 69)( 67, 70)( 74, 80)( 75, 78)( 76, 79)( 83, 89)( 84, 87)( 85, 88)( 92, 98)( 93, 96)( 94, 97)(101,107)(102,105)(103,106);; 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*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(108)!( 1, 55)( 2, 56)( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108); s1 := Sym(108)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)( 44, 51)( 45, 50)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64,100)( 65,102)( 66,101)( 67,106)( 68,108)( 69,107)( 70,103)( 71,105)( 72,104)( 73, 91)( 74, 93)( 75, 92)( 76, 97)( 77, 99)( 78, 98)( 79, 94)( 80, 96)( 81, 95); s2 := Sym(108)!( 1, 11)( 2, 10)( 3, 12)( 4, 15)( 5, 14)( 6, 13)( 7, 16)( 8, 18)( 9, 17)( 19, 20)( 22, 24)( 26, 27)( 28, 38)( 29, 37)( 30, 39)( 31, 42)( 32, 41)( 33, 40)( 34, 43)( 35, 45)( 36, 44)( 46, 47)( 49, 51)( 53, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 69)( 59, 68)( 60, 67)( 61, 70)( 62, 72)( 63, 71)( 73, 74)( 76, 78)( 80, 81)( 82, 92)( 83, 91)( 84, 93)( 85, 96)( 86, 95)( 87, 94)( 88, 97)( 89, 99)( 90, 98)(100,101)(103,105)(107,108); s3 := Sym(108)!( 2, 8)( 3, 6)( 4, 7)( 11, 17)( 12, 15)( 13, 16)( 20, 26)( 21, 24)( 22, 25)( 29, 35)( 30, 33)( 31, 34)( 38, 44)( 39, 42)( 40, 43)( 47, 53)( 48, 51)( 49, 52)( 56, 62)( 57, 60)( 58, 61)( 65, 71)( 66, 69)( 67, 70)( 74, 80)( 75, 78)( 76, 79)( 83, 89)( 84, 87)( 85, 88)( 92, 98)( 93, 96)( 94, 97)(101,107)(102,105)(103,106); 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*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2 >;
References
None.
to this polytope.