Overview
- Group
- SmallGroup(1728,15957)
- Rank
- 4
- Schläfli Type
- {8,18,6}
- Vertices, edges, …
- 8, 72, 54, 6
- Order of s0s1s2s3
- 72
- Order of s0s1s2s3s2s1
- 2
- Also known as
- if this polytope has a name.
Special Properties
- Universal
- Orientable
- Flat
Quotients maximal quotients in bold
2-fold
3-fold
4-fold
6-fold
8-fold
9-fold
12-fold
18-fold
24-fold
27-fold
36-fold
54-fold
72-fold
108-fold
Covers minimal covers in bold
None in this atlas.
Irregular Quotients of which this is a minimal cover
None.
Representations
Permutation Representation (GAP)
s0 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);; s1 := ( 1,109)( 2,111)( 3,110)( 4,115)( 5,117)( 6,116)( 7,112)( 8,114)( 9,113)( 10,129)( 11,128)( 12,127)( 13,135)( 14,134)( 15,133)( 16,132)( 17,131)( 18,130)( 19,120)( 20,119)( 21,118)( 22,126)( 23,125)( 24,124)( 25,123)( 26,122)( 27,121)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)( 33,143)( 34,139)( 35,141)( 36,140)( 37,156)( 38,155)( 39,154)( 40,162)( 41,161)( 42,160)( 43,159)( 44,158)( 45,157)( 46,147)( 47,146)( 48,145)( 49,153)( 50,152)( 51,151)( 52,150)( 53,149)( 54,148)( 55,190)( 56,192)( 57,191)( 58,196)( 59,198)( 60,197)( 61,193)( 62,195)( 63,194)( 64,210)( 65,209)( 66,208)( 67,216)( 68,215)( 69,214)( 70,213)( 71,212)( 72,211)( 73,201)( 74,200)( 75,199)( 76,207)( 77,206)( 78,205)( 79,204)( 80,203)( 81,202)( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)( 89,168)( 90,167)( 91,183)( 92,182)( 93,181)( 94,189)( 95,188)( 96,187)( 97,186)( 98,185)( 99,184)(100,174)(101,173)(102,172)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175);; s2 := ( 1, 13)( 2, 15)( 3, 14)( 4, 10)( 5, 12)( 6, 11)( 7, 16)( 8, 18)( 9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)( 47, 50)( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)(100,105)(101,104)(102,103)(106,108)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,132)(128,131)(129,130)(133,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)(143,153)(144,152)(154,159)(155,158)(156,157)(160,162)(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,186)(182,185)(183,184)(187,189)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(196,205)(197,207)(198,206)(208,213)(209,212)(210,211)(214,216);; s3 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);; poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP)
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(216)!( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216); s1 := Sym(216)!( 1,109)( 2,111)( 3,110)( 4,115)( 5,117)( 6,116)( 7,112)( 8,114)( 9,113)( 10,129)( 11,128)( 12,127)( 13,135)( 14,134)( 15,133)( 16,132)( 17,131)( 18,130)( 19,120)( 20,119)( 21,118)( 22,126)( 23,125)( 24,124)( 25,123)( 26,122)( 27,121)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)( 33,143)( 34,139)( 35,141)( 36,140)( 37,156)( 38,155)( 39,154)( 40,162)( 41,161)( 42,160)( 43,159)( 44,158)( 45,157)( 46,147)( 47,146)( 48,145)( 49,153)( 50,152)( 51,151)( 52,150)( 53,149)( 54,148)( 55,190)( 56,192)( 57,191)( 58,196)( 59,198)( 60,197)( 61,193)( 62,195)( 63,194)( 64,210)( 65,209)( 66,208)( 67,216)( 68,215)( 69,214)( 70,213)( 71,212)( 72,211)( 73,201)( 74,200)( 75,199)( 76,207)( 77,206)( 78,205)( 79,204)( 80,203)( 81,202)( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)( 89,168)( 90,167)( 91,183)( 92,182)( 93,181)( 94,189)( 95,188)( 96,187)( 97,186)( 98,185)( 99,184)(100,174)(101,173)(102,172)(103,180)(104,179)(105,178)(106,177)(107,176)(108,175); s2 := Sym(216)!( 1, 13)( 2, 15)( 3, 14)( 4, 10)( 5, 12)( 6, 11)( 7, 16)( 8, 18)( 9, 17)( 19, 24)( 20, 23)( 21, 22)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 37)( 32, 39)( 33, 38)( 34, 43)( 35, 45)( 36, 44)( 46, 51)( 47, 50)( 48, 49)( 52, 54)( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 61, 70)( 62, 72)( 63, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85, 91)( 86, 93)( 87, 92)( 88, 97)( 89, 99)( 90, 98)(100,105)(101,104)(102,103)(106,108)(109,121)(110,123)(111,122)(112,118)(113,120)(114,119)(115,124)(116,126)(117,125)(127,132)(128,131)(129,130)(133,135)(136,148)(137,150)(138,149)(139,145)(140,147)(141,146)(142,151)(143,153)(144,152)(154,159)(155,158)(156,157)(160,162)(163,175)(164,177)(165,176)(166,172)(167,174)(168,173)(169,178)(170,180)(171,179)(181,186)(182,185)(183,184)(187,189)(190,202)(191,204)(192,203)(193,199)(194,201)(195,200)(196,205)(197,207)(198,206)(208,213)(209,212)(210,211)(214,216); s3 := Sym(216)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216); poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma)
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References
None.
to this polytope.