Overview
- Group
- SmallGroup(1728,12713)
- Rank
- 3
- Schläfli Type
- {24,4}
- Vertices, edges, …
- 216, 432, 36
- Order of s0s1s2
- 24
- Order of s0s1s2s1
- 12
- 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
8-fold
12-fold
24-fold
27-fold
54-fold
108-fold
216-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=<s0*s1*(s2*s1*s0)^2*s1*s2*s1*s0*s2*s1> of order 3
16 facets
72 vertex figures
- 72 of {4}*8
Representations
Permutation Representation (GAP)
s0 := ( 1,136)( 2,137)( 3,138)( 4,144)( 5,142)( 6,143)( 7,140)( 8,141)( 9,139)( 10,156)( 11,154)( 12,155)( 13,161)( 14,162)( 15,160)( 16,157)( 17,158)( 18,159)( 19,146)( 20,147)( 21,145)( 22,151)( 23,152)( 24,153)( 25,150)( 26,148)( 27,149)( 28,109)( 29,110)( 30,111)( 31,117)( 32,115)( 33,116)( 34,113)( 35,114)( 36,112)( 37,129)( 38,127)( 39,128)( 40,134)( 41,135)( 42,133)( 43,130)( 44,131)( 45,132)( 46,119)( 47,120)( 48,118)( 49,124)( 50,125)( 51,126)( 52,123)( 53,121)( 54,122)( 55,163)( 56,164)( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)( 64,183)( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)( 72,186)( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)( 80,175)( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)( 88,194)( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)( 96,214)( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)(104,206)(105,207)(106,204)(107,202)(108,203);; s1 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)( 63, 89)( 64, 99)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)( 70, 93)( 71, 92)( 72, 91)( 73,102)( 74,101)( 75,100)( 76,108)( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,193)(110,195)(111,194)(112,190)(113,192)(114,191)(115,196)(116,198)(117,197)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,210)(128,209)(129,208)(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)(136,166)(137,168)(138,167)(139,163)(140,165)(141,164)(142,169)(143,171)(144,170)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(154,183)(155,182)(156,181)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184);; s2 := ( 1, 55)( 2, 57)( 3, 56)( 4, 70)( 5, 72)( 6, 71)( 7, 78)( 8, 77)( 9, 76)( 10, 65)( 11, 64)( 12, 66)( 13, 80)( 14, 79)( 15, 81)( 16, 58)( 17, 60)( 18, 59)( 19, 75)( 20, 74)( 21, 73)( 22, 63)( 23, 62)( 24, 61)( 25, 68)( 26, 67)( 27, 69)( 28, 82)( 29, 84)( 30, 83)( 31, 97)( 32, 99)( 33, 98)( 34,105)( 35,104)( 36,103)( 37, 92)( 38, 91)( 39, 93)( 40,107)( 41,106)( 42,108)( 43, 85)( 44, 87)( 45, 86)( 46,102)( 47,101)( 48,100)( 49, 90)( 50, 89)( 51, 88)( 52, 95)( 53, 94)( 54, 96)(109,190)(110,192)(111,191)(112,205)(113,207)(114,206)(115,213)(116,212)(117,211)(118,200)(119,199)(120,201)(121,215)(122,214)(123,216)(124,193)(125,195)(126,194)(127,210)(128,209)(129,208)(130,198)(131,197)(132,196)(133,203)(134,202)(135,204)(136,163)(137,165)(138,164)(139,178)(140,180)(141,179)(142,186)(143,185)(144,184)(145,173)(146,172)(147,174)(148,188)(149,187)(150,189)(151,166)(152,168)(153,167)(154,183)(155,182)(156,181)(157,171)(158,170)(159,169)(160,176)(161,175)(162,177);; 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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(216)!( 1,136)( 2,137)( 3,138)( 4,144)( 5,142)( 6,143)( 7,140)( 8,141)( 9,139)( 10,156)( 11,154)( 12,155)( 13,161)( 14,162)( 15,160)( 16,157)( 17,158)( 18,159)( 19,146)( 20,147)( 21,145)( 22,151)( 23,152)( 24,153)( 25,150)( 26,148)( 27,149)( 28,109)( 29,110)( 30,111)( 31,117)( 32,115)( 33,116)( 34,113)( 35,114)( 36,112)( 37,129)( 38,127)( 39,128)( 40,134)( 41,135)( 42,133)( 43,130)( 44,131)( 45,132)( 46,119)( 47,120)( 48,118)( 49,124)( 50,125)( 51,126)( 52,123)( 53,121)( 54,122)( 55,163)( 56,164)( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)( 64,183)( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)( 72,186)( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)( 80,175)( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)( 88,194)( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)( 96,214)( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)(104,206)(105,207)(106,204)(107,202)(108,203); s1 := Sym(216)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)( 63, 89)( 64, 99)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)( 70, 93)( 71, 92)( 72, 91)( 73,102)( 74,101)( 75,100)( 76,108)( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,193)(110,195)(111,194)(112,190)(113,192)(114,191)(115,196)(116,198)(117,197)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)(125,200)(126,199)(127,210)(128,209)(129,208)(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)(136,166)(137,168)(138,167)(139,163)(140,165)(141,164)(142,169)(143,171)(144,170)(145,180)(146,179)(147,178)(148,177)(149,176)(150,175)(151,174)(152,173)(153,172)(154,183)(155,182)(156,181)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184); s2 := Sym(216)!( 1, 55)( 2, 57)( 3, 56)( 4, 70)( 5, 72)( 6, 71)( 7, 78)( 8, 77)( 9, 76)( 10, 65)( 11, 64)( 12, 66)( 13, 80)( 14, 79)( 15, 81)( 16, 58)( 17, 60)( 18, 59)( 19, 75)( 20, 74)( 21, 73)( 22, 63)( 23, 62)( 24, 61)( 25, 68)( 26, 67)( 27, 69)( 28, 82)( 29, 84)( 30, 83)( 31, 97)( 32, 99)( 33, 98)( 34,105)( 35,104)( 36,103)( 37, 92)( 38, 91)( 39, 93)( 40,107)( 41,106)( 42,108)( 43, 85)( 44, 87)( 45, 86)( 46,102)( 47,101)( 48,100)( 49, 90)( 50, 89)( 51, 88)( 52, 95)( 53, 94)( 54, 96)(109,190)(110,192)(111,191)(112,205)(113,207)(114,206)(115,213)(116,212)(117,211)(118,200)(119,199)(120,201)(121,215)(122,214)(123,216)(124,193)(125,195)(126,194)(127,210)(128,209)(129,208)(130,198)(131,197)(132,196)(133,203)(134,202)(135,204)(136,163)(137,165)(138,164)(139,178)(140,180)(141,179)(142,186)(143,185)(144,184)(145,173)(146,172)(147,174)(148,188)(149,187)(150,189)(151,166)(152,168)(153,167)(154,183)(155,182)(156,181)(157,171)(158,170)(159,169)(160,176)(161,175)(162,177); 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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.