Overview
- Group
- SmallGroup(1728,12630)
- Rank
- 3
- Schläfli Type
- {4,12}
- Vertices, edges, …
- 72, 432, 216
- Order of s0s1s2
- 12
- 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)^5*s0*s1*s2> of order 2
108 facets
- 108 of {4}*8
36 vertex figures
- 36 of {12}*24
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s2*s1)^2> of order 3
72 facets
- 72 of {4}*8
24 vertex figures
- 24 of {12}*24
P/N, where N=<(s0*s1*s2*s1)^2*s0*(s2*s1)^2*s0*(s1*s2)^2> of order 3
72 facets
- 72 of {4}*8
24 vertex figures
- 24 of {12}*24
P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^5*s0*s1*s2> of order 4
57 facets
18 vertex figures
- 18 of {12}*24
P/N, where N=<s1*(s2*s1*s0)^4*s2*s1*s2, s0*s1*(s2*s1*s0)^5*s2*s1> of order 4
54 facets
- 54 of {4}*8
19 vertex figures
P/N, where N=<s0*(s2*s1*s0*s1)^2*s2*s1> of order 4
54 facets
- 54 of {4}*8
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*s2*s1)^6, (s0*s1)^2*(s2*s1*s0)^4*s2*s1*s2> of order 4
54 facets
- 54 of {4}*8
18 vertex figures
- 18 of {12}*24
P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^5*s2> of order 4
57 facets
19 vertex figures
P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^3*s1, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 6
36 facets
- 36 of {4}*8
14 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s2*s1)^2, (s0*s2*s1)^6> of order 6
36 facets
- 36 of {4}*8
12 vertex figures
- 12 of {12}*24
P/N, where N=<(s0*s1)^2*s2*s1*s0*(s2*s1)^2*s0*s1*s2> of order 6
36 facets
- 36 of {4}*8
12 vertex figures
- 12 of {12}*24
P/N, where N=<s0*s1*s2*s1*s0*(s1*s2)^3*s1, (s1*s2)^2*(s1*s0*s1*s2)^2> of order 6
36 facets
- 36 of {4}*8
14 vertex figures
P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s2*s1)^2, (s0*s2*s1)^5*s0*s1*s2*s1> of order 6
36 facets
- 36 of {4}*8
12 vertex figures
- 12 of {12}*24
P/N, where N=<(s0*s1)^2, (s2*s1*s0)^2*(s1*s2)^2> of order 6
42 facets
12 vertex figures
- 12 of {12}*24
P/N, where N=<(s0*s1)^2, (s1*s2)^2*(s1*s0*s1*s2)^2> of order 12
21 facets
7 vertex figures
P/N, where N=<(s0*s1)^2, (s2*s1*s0)^2*(s1*s2)^2, s0*s1*(s2*s1*s0)^5*s2> of order 12
21 facets
7 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 2, 3)( 4, 22)( 5, 24)( 6, 23)( 7, 18)( 8, 17)( 9, 16)( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 25, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)( 34, 45)( 35, 44)( 36, 43)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 52, 54)( 55, 82)( 56, 84)( 57, 83)( 58,103)( 59,105)( 60,104)( 61, 99)( 62, 98)( 63, 97)( 64,100)( 65,102)( 66,101)( 67, 94)( 68, 96)( 69, 95)( 70, 90)( 71, 89)( 72, 88)( 73, 91)( 74, 93)( 75, 92)( 76, 85)( 77, 87)( 78, 86)( 79,108)( 80,107)( 81,106)(110,111)(112,130)(113,132)(114,131)(115,126)(116,125)(117,124)(118,127)(119,129)(120,128)(122,123)(133,135)(137,138)(139,157)(140,159)(141,158)(142,153)(143,152)(144,151)(145,154)(146,156)(147,155)(149,150)(160,162)(163,190)(164,192)(165,191)(166,211)(167,213)(168,212)(169,207)(170,206)(171,205)(172,208)(173,210)(174,209)(175,202)(176,204)(177,203)(178,198)(179,197)(180,196)(181,199)(182,201)(183,200)(184,193)(185,195)(186,194)(187,216)(188,215)(189,214);; 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)( 56, 57)( 58, 59)( 61, 63)( 64, 79)( 65, 81)( 66, 80)( 67, 74)( 68, 73)( 69, 75)( 70, 78)( 71, 77)( 72, 76)( 83, 84)( 85, 86)( 88, 90)( 91,106)( 92,108)( 93,107)( 94,101)( 95,100)( 96,102)( 97,105)( 98,104)( 99,103)(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 := ( 1,148)( 2,150)( 3,149)( 4,156)( 5,155)( 6,154)( 7,142)( 8,144)( 9,143)( 10,158)( 11,157)( 12,159)( 13,136)( 14,138)( 15,137)( 16,152)( 17,151)( 18,153)( 19,141)( 20,140)( 21,139)( 22,146)( 23,145)( 24,147)( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,129)( 32,128)( 33,127)( 34,115)( 35,117)( 36,116)( 37,131)( 38,130)( 39,132)( 40,109)( 41,111)( 42,110)( 43,125)( 44,124)( 45,126)( 46,114)( 47,113)( 48,112)( 49,119)( 50,118)( 51,120)( 52,135)( 53,134)( 54,133)( 55,175)( 56,177)( 57,176)( 58,183)( 59,182)( 60,181)( 61,169)( 62,171)( 63,170)( 64,185)( 65,184)( 66,186)( 67,163)( 68,165)( 69,164)( 70,179)( 71,178)( 72,180)( 73,168)( 74,167)( 75,166)( 76,173)( 77,172)( 78,174)( 79,189)( 80,188)( 81,187)( 82,202)( 83,204)( 84,203)( 85,210)( 86,209)( 87,208)( 88,196)( 89,198)( 90,197)( 91,212)( 92,211)( 93,213)( 94,190)( 95,192)( 96,191)( 97,206)( 98,205)( 99,207)(100,195)(101,194)(102,193)(103,200)(104,199)(105,201)(106,216)(107,215)(108,214);; 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,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(216)!( 2, 3)( 4, 22)( 5, 24)( 6, 23)( 7, 18)( 8, 17)( 9, 16)( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 25, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)( 34, 45)( 35, 44)( 36, 43)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 52, 54)( 55, 82)( 56, 84)( 57, 83)( 58,103)( 59,105)( 60,104)( 61, 99)( 62, 98)( 63, 97)( 64,100)( 65,102)( 66,101)( 67, 94)( 68, 96)( 69, 95)( 70, 90)( 71, 89)( 72, 88)( 73, 91)( 74, 93)( 75, 92)( 76, 85)( 77, 87)( 78, 86)( 79,108)( 80,107)( 81,106)(110,111)(112,130)(113,132)(114,131)(115,126)(116,125)(117,124)(118,127)(119,129)(120,128)(122,123)(133,135)(137,138)(139,157)(140,159)(141,158)(142,153)(143,152)(144,151)(145,154)(146,156)(147,155)(149,150)(160,162)(163,190)(164,192)(165,191)(166,211)(167,213)(168,212)(169,207)(170,206)(171,205)(172,208)(173,210)(174,209)(175,202)(176,204)(177,203)(178,198)(179,197)(180,196)(181,199)(182,201)(183,200)(184,193)(185,195)(186,194)(187,216)(188,215)(189,214); 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)( 56, 57)( 58, 59)( 61, 63)( 64, 79)( 65, 81)( 66, 80)( 67, 74)( 68, 73)( 69, 75)( 70, 78)( 71, 77)( 72, 76)( 83, 84)( 85, 86)( 88, 90)( 91,106)( 92,108)( 93,107)( 94,101)( 95,100)( 96,102)( 97,105)( 98,104)( 99,103)(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)!( 1,148)( 2,150)( 3,149)( 4,156)( 5,155)( 6,154)( 7,142)( 8,144)( 9,143)( 10,158)( 11,157)( 12,159)( 13,136)( 14,138)( 15,137)( 16,152)( 17,151)( 18,153)( 19,141)( 20,140)( 21,139)( 22,146)( 23,145)( 24,147)( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,129)( 32,128)( 33,127)( 34,115)( 35,117)( 36,116)( 37,131)( 38,130)( 39,132)( 40,109)( 41,111)( 42,110)( 43,125)( 44,124)( 45,126)( 46,114)( 47,113)( 48,112)( 49,119)( 50,118)( 51,120)( 52,135)( 53,134)( 54,133)( 55,175)( 56,177)( 57,176)( 58,183)( 59,182)( 60,181)( 61,169)( 62,171)( 63,170)( 64,185)( 65,184)( 66,186)( 67,163)( 68,165)( 69,164)( 70,179)( 71,178)( 72,180)( 73,168)( 74,167)( 75,166)( 76,173)( 77,172)( 78,174)( 79,189)( 80,188)( 81,187)( 82,202)( 83,204)( 84,203)( 85,210)( 86,209)( 87,208)( 88,196)( 89,198)( 90,197)( 91,212)( 92,211)( 93,213)( 94,190)( 95,192)( 96,191)( 97,206)( 98,205)( 99,207)(100,195)(101,194)(102,193)(103,200)(104,199)(105,201)(106,216)(107,215)(108,214); 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, 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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.