Overview
- Group
- SmallGroup(1728,46671)
- Rank
- 4
- Schläfli Type
- {4,4,6}
- Vertices, edges, …
- 4, 72, 108, 54
- 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=<(s1*s2)^2> of order 2
30 facets
4 vertex figures
- 4 of 2-fold non-regular quotient of {4,6}*432b
P/N, where N=<s2*s3*s2*s1*(s2*s3)^2*s2*s1*s3*s2*s3> of order 2
27 facets
- 27 of {4,4}*32
4 vertex figures
- 4 of 2-fold non-regular quotient of {4,6}*432b
P/N, where N=<s1*s3*s2*s1*(s2*s3)^2*(s2*s1*s3)^2> of order 2
27 facets
- 27 of {4,4}*32
4 vertex figures
- 4 of 2-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2*(s3*s2*s1)^2> of order 3
18 facets
- 18 of {4,4}*32
4 vertex figures
- 4 of 3-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s2*s1*s2*s3)^2> of order 3
18 facets
- 18 of {4,4}*32
4 vertex figures
- 4 of 3-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2*(s3*s2*s1)^2*(s2*s3)^2> of order 3
18 facets
- 18 of {4,4}*32
4 vertex figures
- 4 of 3-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2*(s3*s2)^2*(s1*s3*s2)^2> of order 3
18 facets
- 18 of {4,4}*32
4 vertex figures
- 4 of 3-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2, s3*s2*s1*(s2*s3)^2*s2*s1*s3*s2*s3> of order 4
15 facets
4 vertex figures
- 4 of 4-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s2*s3)^3, (s1*s2)^2*(s3*s2)^2*s1*s3*s2*s1> of order 6
9 facets
- 9 of {4,4}*32
4 vertex figures
- 4 of 6-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2, (s3*s2*s1)^2*(s2*s3)^2> of order 6
12 facets
4 vertex figures
- 4 of 6-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2, s1*s3*s2*s1*s2*s3> of order 6
12 facets
4 vertex figures
- 4 of 6-fold non-regular quotient of {4,6}*432b
P/N, where N=<s1*s2*s3*s2*s1*s3, (s2*s1*s2*s3)^2> of order 9
6 facets
- 6 of {4,4}*32
4 vertex figures
- 4 of 9-fold non-regular quotient of {4,6}*432b
P/N, where N=<s1*s2*s3*s2*s1*s3, s2*s1*(s3*s2)^2*s1*s3*s2*s3> of order 9
6 facets
- 6 of {4,4}*32
4 vertex figures
- 4 of 9-fold non-regular quotient of {4,6}*432b
P/N, where N=<(s1*s2)^2, s1*(s2*s3)^2*s2*s1*s3> of order 12
6 facets
4 vertex figures
- 4 of 12-fold non-regular quotient of {4,6}*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 := ( 4, 9)( 5, 7)( 6, 8)( 13, 18)( 14, 16)( 15, 17)( 22, 27)( 23, 25)( 24, 26)( 31, 36)( 32, 34)( 33, 35)( 40, 45)( 41, 43)( 42, 44)( 49, 54)( 50, 52)( 51, 53)( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)( 60, 89)( 61, 86)( 62, 87)( 63, 85)( 64, 91)( 65, 92)( 66, 93)( 67, 99)( 68, 97)( 69, 98)( 70, 95)( 71, 96)( 72, 94)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)( 79,104)( 80,105)( 81,103);; s2 := ( 2, 8)( 3, 6)( 4, 7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)( 14, 23)( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)( 37, 46)( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)( 45, 54)( 56, 62)( 57, 60)( 58, 61)( 64, 73)( 65, 80)( 66, 78)( 67, 79)( 68, 77)( 69, 75)( 70, 76)( 71, 74)( 72, 81)( 83, 89)( 84, 87)( 85, 88)( 91,100)( 92,107)( 93,105)( 94,106)( 95,104)( 96,102)( 97,103)( 98,101)( 99,108);; s3 := ( 1, 11)( 2, 10)( 3, 12)( 4, 17)( 5, 16)( 6, 18)( 7, 14)( 8, 13)( 9, 15)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 38)( 29, 37)( 30, 39)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 47)( 49, 53)( 50, 52)( 51, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 71)( 59, 70)( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 92)( 83, 91)( 84, 93)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)(100,101)(103,107)(104,106)(105,108);; 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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*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)!( 4, 9)( 5, 7)( 6, 8)( 13, 18)( 14, 16)( 15, 17)( 22, 27)( 23, 25)( 24, 26)( 31, 36)( 32, 34)( 33, 35)( 40, 45)( 41, 43)( 42, 44)( 49, 54)( 50, 52)( 51, 53)( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)( 60, 89)( 61, 86)( 62, 87)( 63, 85)( 64, 91)( 65, 92)( 66, 93)( 67, 99)( 68, 97)( 69, 98)( 70, 95)( 71, 96)( 72, 94)( 73,100)( 74,101)( 75,102)( 76,108)( 77,106)( 78,107)( 79,104)( 80,105)( 81,103); s2 := Sym(108)!( 2, 8)( 3, 6)( 4, 7)( 10, 19)( 11, 26)( 12, 24)( 13, 25)( 14, 23)( 15, 21)( 16, 22)( 17, 20)( 18, 27)( 29, 35)( 30, 33)( 31, 34)( 37, 46)( 38, 53)( 39, 51)( 40, 52)( 41, 50)( 42, 48)( 43, 49)( 44, 47)( 45, 54)( 56, 62)( 57, 60)( 58, 61)( 64, 73)( 65, 80)( 66, 78)( 67, 79)( 68, 77)( 69, 75)( 70, 76)( 71, 74)( 72, 81)( 83, 89)( 84, 87)( 85, 88)( 91,100)( 92,107)( 93,105)( 94,106)( 95,104)( 96,102)( 97,103)( 98,101)( 99,108); s3 := Sym(108)!( 1, 11)( 2, 10)( 3, 12)( 4, 17)( 5, 16)( 6, 18)( 7, 14)( 8, 13)( 9, 15)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 38)( 29, 37)( 30, 39)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 46, 47)( 49, 53)( 50, 52)( 51, 54)( 55, 65)( 56, 64)( 57, 66)( 58, 71)( 59, 70)( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 92)( 83, 91)( 84, 93)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)(100,101)(103,107)(104,106)(105,108); 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, s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2 >;
References
None.
to this polytope.