Overview
- Group
- SmallGroup(1728,12703)
- Rank
- 3
- Schläfli Type
- {8,12}
- Vertices, edges, …
- 72, 432, 108
- Order of s0s1s2
- 24
- 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
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*(s2*s1)^2*(s0*s1)^2*s2*s1*s2> of order 2
54 facets
- 54 of {8}*16
36 vertex figures
- 36 of {12}*24
P/N, where N=<s1*(s2*s1*s0)^2*(s2*s1)^2*(s0*s1)^2*s2*s1*s2> of order 2
54 facets
- 54 of {8}*16
40 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s2*s1)^2> of order 3
36 facets
- 36 of {8}*16
24 vertex figures
- 24 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2> of order 3
36 facets
- 36 of {8}*16
24 vertex figures
- 24 of {12}*24
P/N, where N=<s0*(s1*s2)^3*s1*s0*s1*s2*s1, (s0*s1)^2*s2*s1*s0*(s1*s2)^2*s1*s0*s2> of order 6
18 facets
- 18 of {8}*16
16 vertex figures
P/N, where N=<(s0*s1*s2*s1)^2, (s1*s2)^2*(s1*s0*s1*s2)^2> of order 6
18 facets
- 18 of {8}*16
16 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s2, s1*s0*s1*s2*s1*s0*(s2*s1)^2*s0*s1*s2> of order 6
18 facets
- 18 of {8}*16
12 vertex figures
- 12 of {12}*24
Representations
Permutation Representation (GAP)
s0 := ( 1,109)( 2,111)( 3,110)( 4,130)( 5,132)( 6,131)( 7,126)( 8,125)( 9,124)( 10,127)( 11,129)( 12,128)( 13,121)( 14,123)( 15,122)( 16,117)( 17,116)( 18,115)( 19,118)( 20,120)( 21,119)( 22,112)( 23,114)( 24,113)( 25,135)( 26,134)( 27,133)( 28,136)( 29,138)( 30,137)( 31,157)( 32,159)( 33,158)( 34,153)( 35,152)( 36,151)( 37,154)( 38,156)( 39,155)( 40,148)( 41,150)( 42,149)( 43,144)( 44,143)( 45,142)( 46,145)( 47,147)( 48,146)( 49,139)( 50,141)( 51,140)( 52,162)( 53,161)( 54,160)( 55,190)( 56,192)( 57,191)( 58,211)( 59,213)( 60,212)( 61,207)( 62,206)( 63,205)( 64,208)( 65,210)( 66,209)( 67,202)( 68,204)( 69,203)( 70,198)( 71,197)( 72,196)( 73,199)( 74,201)( 75,200)( 76,193)( 77,195)( 78,194)( 79,216)( 80,215)( 81,214)( 82,163)( 83,165)( 84,164)( 85,184)( 86,186)( 87,185)( 88,180)( 89,179)( 90,178)( 91,181)( 92,183)( 93,182)( 94,175)( 95,177)( 96,176)( 97,171)( 98,170)( 99,169)(100,172)(101,174)(102,173)(103,166)(104,168)(105,167)(106,189)(107,188)(108,187);; s1 := ( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)( 79, 91)( 80, 93)( 81, 92)(109,163)(110,165)(111,164)(112,167)(113,166)(114,168)(115,171)(116,170)(117,169)(118,187)(119,189)(120,188)(121,182)(122,181)(123,183)(124,186)(125,185)(126,184)(127,176)(128,175)(129,177)(130,180)(131,179)(132,178)(133,172)(134,174)(135,173)(136,190)(137,192)(138,191)(139,194)(140,193)(141,195)(142,198)(143,197)(144,196)(145,214)(146,216)(147,215)(148,209)(149,208)(150,210)(151,213)(152,212)(153,211)(154,203)(155,202)(156,204)(157,207)(158,206)(159,205)(160,199)(161,201)(162,200);; s2 := ( 1, 13)( 2, 15)( 3, 14)( 4, 21)( 5, 20)( 6, 19)( 8, 9)( 10, 23)( 11, 22)( 12, 24)( 16, 17)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 48)( 32, 47)( 33, 46)( 35, 36)( 37, 50)( 38, 49)( 39, 51)( 43, 44)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 75)( 59, 74)( 60, 73)( 62, 63)( 64, 77)( 65, 76)( 66, 78)( 70, 71)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85,102)( 86,101)( 87,100)( 89, 90)( 91,104)( 92,103)( 93,105)( 97, 98)(106,108)(109,121)(110,123)(111,122)(112,129)(113,128)(114,127)(116,117)(118,131)(119,130)(120,132)(124,125)(133,135)(136,148)(137,150)(138,149)(139,156)(140,155)(141,154)(143,144)(145,158)(146,157)(147,159)(151,152)(160,162)(163,175)(164,177)(165,176)(166,183)(167,182)(168,181)(170,171)(172,185)(173,184)(174,186)(178,179)(187,189)(190,202)(191,204)(192,203)(193,210)(194,209)(195,208)(197,198)(199,212)(200,211)(201,213)(205,206)(214,216);; 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*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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)!( 1,109)( 2,111)( 3,110)( 4,130)( 5,132)( 6,131)( 7,126)( 8,125)( 9,124)( 10,127)( 11,129)( 12,128)( 13,121)( 14,123)( 15,122)( 16,117)( 17,116)( 18,115)( 19,118)( 20,120)( 21,119)( 22,112)( 23,114)( 24,113)( 25,135)( 26,134)( 27,133)( 28,136)( 29,138)( 30,137)( 31,157)( 32,159)( 33,158)( 34,153)( 35,152)( 36,151)( 37,154)( 38,156)( 39,155)( 40,148)( 41,150)( 42,149)( 43,144)( 44,143)( 45,142)( 46,145)( 47,147)( 48,146)( 49,139)( 50,141)( 51,140)( 52,162)( 53,161)( 54,160)( 55,190)( 56,192)( 57,191)( 58,211)( 59,213)( 60,212)( 61,207)( 62,206)( 63,205)( 64,208)( 65,210)( 66,209)( 67,202)( 68,204)( 69,203)( 70,198)( 71,197)( 72,196)( 73,199)( 74,201)( 75,200)( 76,193)( 77,195)( 78,194)( 79,216)( 80,215)( 81,214)( 82,163)( 83,165)( 84,164)( 85,184)( 86,186)( 87,185)( 88,180)( 89,179)( 90,178)( 91,181)( 92,183)( 93,182)( 94,175)( 95,177)( 96,176)( 97,171)( 98,170)( 99,169)(100,172)(101,174)(102,173)(103,166)(104,168)(105,167)(106,189)(107,188)(108,187); s1 := Sym(216)!( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)( 79, 91)( 80, 93)( 81, 92)(109,163)(110,165)(111,164)(112,167)(113,166)(114,168)(115,171)(116,170)(117,169)(118,187)(119,189)(120,188)(121,182)(122,181)(123,183)(124,186)(125,185)(126,184)(127,176)(128,175)(129,177)(130,180)(131,179)(132,178)(133,172)(134,174)(135,173)(136,190)(137,192)(138,191)(139,194)(140,193)(141,195)(142,198)(143,197)(144,196)(145,214)(146,216)(147,215)(148,209)(149,208)(150,210)(151,213)(152,212)(153,211)(154,203)(155,202)(156,204)(157,207)(158,206)(159,205)(160,199)(161,201)(162,200); s2 := Sym(216)!( 1, 13)( 2, 15)( 3, 14)( 4, 21)( 5, 20)( 6, 19)( 8, 9)( 10, 23)( 11, 22)( 12, 24)( 16, 17)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 48)( 32, 47)( 33, 46)( 35, 36)( 37, 50)( 38, 49)( 39, 51)( 43, 44)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 75)( 59, 74)( 60, 73)( 62, 63)( 64, 77)( 65, 76)( 66, 78)( 70, 71)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85,102)( 86,101)( 87,100)( 89, 90)( 91,104)( 92,103)( 93,105)( 97, 98)(106,108)(109,121)(110,123)(111,122)(112,129)(113,128)(114,127)(116,117)(118,131)(119,130)(120,132)(124,125)(133,135)(136,148)(137,150)(138,149)(139,156)(140,155)(141,154)(143,144)(145,158)(146,157)(147,159)(151,152)(160,162)(163,175)(164,177)(165,176)(166,183)(167,182)(168,181)(170,171)(172,185)(173,184)(174,186)(178,179)(187,189)(190,202)(191,204)(192,203)(193,210)(194,209)(195,208)(197,198)(199,212)(200,211)(201,213)(205,206)(214,216); 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, 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*s0*s1*s0*s1*s2*s1, s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, s0*s1*s2*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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.