Overview
- Group
- SmallGroup(864,4080)
- Rank
- 3
- Schläfli Type
- {12,4}
- Vertices, edges, …
- 108, 216, 36
- 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
6-fold
9-fold
12-fold
18-fold
27-fold
36-fold
54-fold
72-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*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 2
18 facets
- 18 of {12}*24
54 vertex figures
- 54 of {4}*8
P/N, where N=<(s2*s1*s0)^2*s1*(s2*s1*s0)^2*s2*s1*s2> of order 2
18 facets
- 18 of {12}*24
60 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 3
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s1*s0)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 3
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 3
12 facets
- 12 of {12}*24
36 vertex figures
- 36 of {4}*8
P/N, where N=<(s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2, s0*s1*s0*(s2*(s1*s0)^2)^2*s2*s1> of order 4
9 facets
- 9 of {12}*24
30 vertex figures
P/N, where N=<(s0*s1)^6, (s0*s2*s1)^2*s0*s1*(s2*s1*s0)^2*s2*s1*s2> of order 4
10 facets
27 vertex figures
- 27 of {4}*8
P/N, where N=<(s0*s1*s2*s1)^3, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 4
9 facets
- 9 of {12}*24
27 vertex figures
- 27 of {4}*8
P/N, where N=<(s0*s1*s2*s1)^2, ((s1*s0)^2*s2)^2*s1*s0*s2*s1*s2> of order 6
6 facets
- 6 of {12}*24
18 vertex figures
- 18 of {4}*8
P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 6
6 facets
- 6 of {12}*24
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1*s2*s1)^3> of order 6
6 facets
- 6 of {12}*24
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1)^6, (s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 6
8 facets
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s0*s2*(s1*s0)^2*s2*s1*s0*s1*s2> of order 6
6 facets
- 6 of {12}*24
18 vertex figures
- 18 of {4}*8
P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s1*s0*s1*s2)^2*s1*s0*s2*s1*s2> of order 6
6 facets
- 6 of {12}*24
18 vertex figures
- 18 of {4}*8
P/N, where N=<(s0*s1*s2*s1)^2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6
6 facets
- 6 of {12}*24
24 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 22)( 5, 24)( 6, 23)( 7, 16)( 8, 18)( 9, 17)( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 26, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)( 34, 43)( 35, 45)( 36, 44)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 53, 54)( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 70)( 62, 72)( 63, 71)( 64, 73)( 65, 75)( 66, 74)( 68, 69)( 80, 81)( 83, 84)( 85,103)( 86,105)( 87,104)( 88, 97)( 89, 99)( 90, 98)( 91,100)( 92,102)( 93,101)( 95, 96)(107,108)(109,190)(110,192)(111,191)(112,211)(113,213)(114,212)(115,205)(116,207)(117,206)(118,208)(119,210)(120,209)(121,202)(122,204)(123,203)(124,196)(125,198)(126,197)(127,199)(128,201)(129,200)(130,193)(131,195)(132,194)(133,214)(134,216)(135,215)(136,163)(137,165)(138,164)(139,184)(140,186)(141,185)(142,178)(143,180)(144,179)(145,181)(146,183)(147,182)(148,175)(149,177)(150,176)(151,169)(152,171)(153,170)(154,172)(155,174)(156,173)(157,166)(158,168)(159,167)(160,187)(161,189)(162,188);; s1 := ( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)( 9,117)( 10,134)( 11,133)( 12,135)( 13,128)( 14,127)( 15,129)( 16,131)( 17,130)( 18,132)( 19,122)( 20,121)( 21,123)( 22,125)( 23,124)( 24,126)( 25,119)( 26,118)( 27,120)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,161)( 38,160)( 39,162)( 40,155)( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,149)( 47,148)( 48,150)( 49,152)( 50,151)( 51,153)( 52,146)( 53,145)( 54,147)( 55,164)( 56,163)( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)( 64,188)( 65,187)( 66,189)( 67,182)( 68,181)( 69,183)( 70,185)( 71,184)( 72,186)( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,173)( 80,172)( 81,174)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)( 88,197)( 89,196)( 90,198)( 91,215)( 92,214)( 93,216)( 94,209)( 95,208)( 96,210)( 97,212)( 98,211)( 99,213)(100,203)(101,202)(102,204)(103,206)(104,205)(105,207)(106,200)(107,199)(108,201);; s2 := ( 1, 67)( 2, 68)( 3, 69)( 4, 73)( 5, 74)( 6, 75)( 7, 61)( 8, 62)( 9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 55)( 14, 56)( 15, 57)( 16, 70)( 17, 71)( 18, 72)( 19, 58)( 20, 59)( 21, 60)( 22, 64)( 23, 65)( 24, 66)( 25, 79)( 26, 80)( 27, 81)( 28, 94)( 29, 95)( 30, 96)( 31,100)( 32,101)( 33,102)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)( 40, 82)( 41, 83)( 42, 84)( 43, 97)( 44, 98)( 45, 99)( 46, 85)( 47, 86)( 48, 87)( 49, 91)( 50, 92)( 51, 93)( 52,106)( 53,107)( 54,108)(109,121)(110,122)(111,123)(112,127)(113,128)(114,129)(118,130)(119,131)(120,132)(136,148)(137,149)(138,150)(139,154)(140,155)(141,156)(145,157)(146,158)(147,159)(163,175)(164,176)(165,177)(166,181)(167,182)(168,183)(172,184)(173,185)(174,186)(190,202)(191,203)(192,204)(193,208)(194,209)(195,210)(199,211)(200,212)(201,213);; 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*s0*s1*s0*s1*s2*s1*s0*s1*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*s0*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(216)!( 2, 3)( 4, 22)( 5, 24)( 6, 23)( 7, 16)( 8, 18)( 9, 17)( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 26, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)( 34, 43)( 35, 45)( 36, 44)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 53, 54)( 56, 57)( 58, 76)( 59, 78)( 60, 77)( 61, 70)( 62, 72)( 63, 71)( 64, 73)( 65, 75)( 66, 74)( 68, 69)( 80, 81)( 83, 84)( 85,103)( 86,105)( 87,104)( 88, 97)( 89, 99)( 90, 98)( 91,100)( 92,102)( 93,101)( 95, 96)(107,108)(109,190)(110,192)(111,191)(112,211)(113,213)(114,212)(115,205)(116,207)(117,206)(118,208)(119,210)(120,209)(121,202)(122,204)(123,203)(124,196)(125,198)(126,197)(127,199)(128,201)(129,200)(130,193)(131,195)(132,194)(133,214)(134,216)(135,215)(136,163)(137,165)(138,164)(139,184)(140,186)(141,185)(142,178)(143,180)(144,179)(145,181)(146,183)(147,182)(148,175)(149,177)(150,176)(151,169)(152,171)(153,170)(154,172)(155,174)(156,173)(157,166)(158,168)(159,167)(160,187)(161,189)(162,188); s1 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)( 9,117)( 10,134)( 11,133)( 12,135)( 13,128)( 14,127)( 15,129)( 16,131)( 17,130)( 18,132)( 19,122)( 20,121)( 21,123)( 22,125)( 23,124)( 24,126)( 25,119)( 26,118)( 27,120)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,161)( 38,160)( 39,162)( 40,155)( 41,154)( 42,156)( 43,158)( 44,157)( 45,159)( 46,149)( 47,148)( 48,150)( 49,152)( 50,151)( 51,153)( 52,146)( 53,145)( 54,147)( 55,164)( 56,163)( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)( 64,188)( 65,187)( 66,189)( 67,182)( 68,181)( 69,183)( 70,185)( 71,184)( 72,186)( 73,176)( 74,175)( 75,177)( 76,179)( 77,178)( 78,180)( 79,173)( 80,172)( 81,174)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)( 88,197)( 89,196)( 90,198)( 91,215)( 92,214)( 93,216)( 94,209)( 95,208)( 96,210)( 97,212)( 98,211)( 99,213)(100,203)(101,202)(102,204)(103,206)(104,205)(105,207)(106,200)(107,199)(108,201); s2 := Sym(216)!( 1, 67)( 2, 68)( 3, 69)( 4, 73)( 5, 74)( 6, 75)( 7, 61)( 8, 62)( 9, 63)( 10, 76)( 11, 77)( 12, 78)( 13, 55)( 14, 56)( 15, 57)( 16, 70)( 17, 71)( 18, 72)( 19, 58)( 20, 59)( 21, 60)( 22, 64)( 23, 65)( 24, 66)( 25, 79)( 26, 80)( 27, 81)( 28, 94)( 29, 95)( 30, 96)( 31,100)( 32,101)( 33,102)( 34, 88)( 35, 89)( 36, 90)( 37,103)( 38,104)( 39,105)( 40, 82)( 41, 83)( 42, 84)( 43, 97)( 44, 98)( 45, 99)( 46, 85)( 47, 86)( 48, 87)( 49, 91)( 50, 92)( 51, 93)( 52,106)( 53,107)( 54,108)(109,121)(110,122)(111,123)(112,127)(113,128)(114,129)(118,130)(119,131)(120,132)(136,148)(137,149)(138,150)(139,154)(140,155)(141,156)(145,157)(146,158)(147,159)(163,175)(164,176)(165,177)(166,181)(167,182)(168,183)(172,184)(173,185)(174,186)(190,202)(191,203)(192,204)(193,208)(194,209)(195,210)(199,211)(200,212)(201,213); poly := sub<Sym(216)|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*s0*s1*s0*s1*s2*s1*s0*s1*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*s0*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.