Overview
- Group
- SmallGroup(864,2282)
- Rank
- 3
- Schläfli Type
- {4,12}
- Vertices, edges, …
- 36, 216, 108
- Order of s0s1s2
- 12
- Order of s0s1s2s1
- 6
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Orientable
Quotients maximal quotients in bold
2-fold
3-fold
4-fold
6-fold
12-fold
27-fold
54-fold
108-fold
Covers minimal covers in bold
2-fold
Irregular Quotients of which this is a minimal cover
Click an entry to reveal its facets and vertex figures.
P/N, where N=<(s0*s1)^2*(s2*s1*s0*s1)^2*s2> of order 2
54 facets
- 54 of {4}*8
18 vertex figures
- 18 of {12}*24
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2> of order 3
36 facets
- 36 of {4}*8
12 vertex figures
- 12 of {12}*24
P/N, where N=<s0*(s1*s0*s2)^3*s1*s2> of order 3
36 facets
- 36 of {4}*8
12 vertex figures
- 12 of {12}*24
P/N, where N=<(s0*s1)^2*(s2*s1*s0*s1)^2*s2, s0*s1*s0*(s2*s1*s0*s1)^2*s2*s1> of order 4
27 facets
- 27 of {4}*8
10 vertex figures
P/N, where N=<(s0*s1*s2*s1)^3, (s0*s1)^2*(s2*s1*s0*s1)^2*s2> of order 4
27 facets
- 27 of {4}*8
9 vertex figures
- 9 of {12}*24
P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^4*s2*s1*s2> of order 4
30 facets
9 vertex figures
- 9 of {12}*24
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s2*s1*s0*(s2*s1)^2*s0*s1*s2*s1> of order 6
18 facets
- 18 of {4}*8
6 vertex figures
- 6 of {12}*24
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*s0*(s2*s1*s0*s1)^2*s2*s1> of order 6
18 facets
- 18 of {4}*8
6 vertex figures
- 6 of {12}*24
P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*(s1*s2)^3*s1*s0*s1*s2*s1> of order 6
18 facets
- 18 of {4}*8
8 vertex figures
P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s1*s0*s1*s2*s1*s0*(s1*s2)^3> of order 6
18 facets
- 18 of {4}*8
8 vertex figures
P/N, where N=<(s0*s2*s1)^2*s0*s1*s2*s1> of order 6
18 facets
- 18 of {4}*8
6 vertex figures
- 6 of {12}*24
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 5)( 7, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 32)( 34, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 50)( 41, 49)( 42, 51)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 59)( 61, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 77)( 68, 76)( 69, 78)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 85, 86)( 88, 90)( 91,100)( 92,102)( 93,101)( 94,104)( 95,103)( 96,105)( 97,108)( 98,107)( 99,106);; s1 := ( 2, 3)( 4, 12)( 5, 11)( 6, 10)( 7, 20)( 8, 19)( 9, 21)( 13, 15)( 16, 24)( 17, 23)( 18, 22)( 26, 27)( 29, 30)( 31, 39)( 32, 38)( 33, 37)( 34, 47)( 35, 46)( 36, 48)( 40, 42)( 43, 51)( 44, 50)( 45, 49)( 53, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 93)( 59, 92)( 60, 91)( 61,101)( 62,100)( 63,102)( 64, 87)( 65, 86)( 66, 85)( 67, 96)( 68, 95)( 69, 94)( 70,105)( 71,104)( 72,103)( 73, 89)( 74, 88)( 75, 90)( 76, 99)( 77, 98)( 78, 97)( 79,106)( 80,108)( 81,107);; s2 := ( 1, 62)( 2, 61)( 3, 63)( 4, 59)( 5, 58)( 6, 60)( 7, 56)( 8, 55)( 9, 57)( 10, 71)( 11, 70)( 12, 72)( 13, 68)( 14, 67)( 15, 69)( 16, 65)( 17, 64)( 18, 66)( 19, 80)( 20, 79)( 21, 81)( 22, 77)( 23, 76)( 24, 78)( 25, 74)( 26, 73)( 27, 75)( 28, 89)( 29, 88)( 30, 90)( 31, 86)( 32, 85)( 33, 87)( 34, 83)( 35, 82)( 36, 84)( 37, 98)( 38, 97)( 39, 99)( 40, 95)( 41, 94)( 42, 96)( 43, 92)( 44, 91)( 45, 93)( 46,107)( 47,106)( 48,108)( 49,104)( 50,103)( 51,105)( 52,101)( 53,100)( 54,102);; 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,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*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(108)!( 2, 3)( 4, 5)( 7, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 32)( 34, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 50)( 41, 49)( 42, 51)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 59)( 61, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 77)( 68, 76)( 69, 78)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 85, 86)( 88, 90)( 91,100)( 92,102)( 93,101)( 94,104)( 95,103)( 96,105)( 97,108)( 98,107)( 99,106); s1 := Sym(108)!( 2, 3)( 4, 12)( 5, 11)( 6, 10)( 7, 20)( 8, 19)( 9, 21)( 13, 15)( 16, 24)( 17, 23)( 18, 22)( 26, 27)( 29, 30)( 31, 39)( 32, 38)( 33, 37)( 34, 47)( 35, 46)( 36, 48)( 40, 42)( 43, 51)( 44, 50)( 45, 49)( 53, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 93)( 59, 92)( 60, 91)( 61,101)( 62,100)( 63,102)( 64, 87)( 65, 86)( 66, 85)( 67, 96)( 68, 95)( 69, 94)( 70,105)( 71,104)( 72,103)( 73, 89)( 74, 88)( 75, 90)( 76, 99)( 77, 98)( 78, 97)( 79,106)( 80,108)( 81,107); s2 := Sym(108)!( 1, 62)( 2, 61)( 3, 63)( 4, 59)( 5, 58)( 6, 60)( 7, 56)( 8, 55)( 9, 57)( 10, 71)( 11, 70)( 12, 72)( 13, 68)( 14, 67)( 15, 69)( 16, 65)( 17, 64)( 18, 66)( 19, 80)( 20, 79)( 21, 81)( 22, 77)( 23, 76)( 24, 78)( 25, 74)( 26, 73)( 27, 75)( 28, 89)( 29, 88)( 30, 90)( 31, 86)( 32, 85)( 33, 87)( 34, 83)( 35, 82)( 36, 84)( 37, 98)( 38, 97)( 39, 99)( 40, 95)( 41, 94)( 42, 96)( 43, 92)( 44, 91)( 45, 93)( 46,107)( 47,106)( 48,108)( 49,104)( 50,103)( 51,105)( 52,101)( 53,100)( 54,102); poly := sub<Sym(108)|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, s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*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.